Kurt Wagner likes the backroads. It drives his wife mad, he says, as we pootle through the suburbs of Nashville in his battered Toyota pickup (a stick shift; they're much cheaper secondhand) – she just wants him to go from A to B along the most direct roads. Thing is, he says, the most direct roads aren't always the quickest.

We are braving the traffic to reach Nolensville, 10 miles or so south of the city, where the Tennessee countryside starts to replace the suburban strip malls. His reward for a day of talking about his new album with Lambchop will be baby back ribs – with the Memphis dry rub – from Martin's, the best barbecue place around. "There are barbecue purists who say the meat should stick to the bone," he says, once we've started on the colossal plates of pork. He holds up a rib denuded of all flesh. "They're wrong. This is good, right?"

Barbecued ribs, pickup trucks – it all feeds into the image the British media has constructed of Wagner. He had a job laying wooden floors even after Lambchop became successful! He wears caps advertising agricultural produce! He's from the south! He must be some sort of redneck savant! Why else would he produce these wistful, shy songs, melding lush 70s-style soul to American roots music?

But that's not Wagner. Earlier in the day, in the studio where he paints, he talks about exploring the late 70s Memphis rockabilly scene, about going to art school in Montana, where he became part of the circle of the novelists Thomas McGuane and Richard Brautigan. Later, as we drive around his hometown, he tells me about his dad, a biochemist, and how since a heart attack he's finally started talking about the breakthroughs he made investigating folic acid. He talks about his parents' disappointment when he chose not to follow a reputable career, but instead took up manual labouring to fund his art. It's more Five Easy Pieces than Deliverance.

He seems to live in a very uneasy embrace with Nashville. Lambchop couldn't have existed without it – as the capital of country music, it offers all the infrastructure a band needs to get working for themselves: studios, pressing plants, engineers and so on. But it was a horrible place to grow up. Wagner had long hair – "down to my waist" – and the good ol' boys would throw things at him on the street and shout abuse. There were no rock'n'roll venues, so instead he'd sneak into the bluegrass clubs. "They would let us in when we were 15, 16 years old – if we behaved ourselves they'd let us sit at the back and bring us a pitcher of beer. It turned out it was this amazing music – very fast and thrashlike and exciting. We couldn't go to a country bar – we'd have gotten thrown out just because our hair was too long." Nashville's attitudes "made me want to get the fuck outta here as soon as I could. So I was 17 and I went to Memphis and from there I kept going for a while."

Now, though, in his mid-50s, he couldn't care less about whether British journalists want to portray him as some tobacco-chewing, dog-owning, gun-wielding, share-cropping southerner: "I didn't have a problem with being misconstrued. I was happy to be construed at all," he says, smiling.

Wagner never intended to be a musician. That only happened when he moved back to Nashville in 1986, after Memphis, after Montana, and then a spell living in Chicago. "I truly intended on just being content being a painter," he says. He'd made music all the way through art school, but "never looked at it as an end to my creative thing". It crept up on him as Lambchop slowly transmogrified from complete unknowns to beloved cult through the late 90s.

And even when they started winning a substantial audience, he remained blissfully unaware that he had entered a game where there were rules by which he was meant to play. When Lambchop headlined the Royal Festival Hall in London in 2000, they took to the stage and played their forthcoming album, Is a Woman, instead of Nixon, which they were promoting. "I didn't know that was what we were supposed do and I got scolded: 'Dude, don't you know? That's what they wanna hear.' I was: 'If I was a fan I'd love to hear what they're doing now – I've got the record, I know what that sounds like. I wanna see what's next!' But I didn't realise that's how it works. I guess I figured it out eventually."

Crucial to the story of Lambchop is the late Vic Chesnutt, to whom their 11th album, Mr M, is dedicated. Chesnutt was a Georgia-based singer-songwriter who took – like Lambchop – an off-kilter approach to American roots music across scores of albums and collaborations. "He was pivotal," Wagner says. Soon after Wagner returned to Nashville from Chicago, he went to see Chesnutt play. "I didn't know what time the show was – the club was in some place I'd never heard of, so I went down early to try and find out where it was. He'd just finished his soundcheck and was hanging out and so we started talking and we hit it off pretty well. And over the years he would keep coming through town and I finally revealed to him that I was making music and played him some little tapes I'd done, and he was very encouraging. We finally got around to making a record– I guess he'd been over in Europe doing a press tour or something, and he started talking to anyone who would listen about this crazy dude over in Nashville and his weird little band, and I sort of attribute the interest we got over there to begin with to him even letting people know we existed."

Chesnutt had been injured in a car accident in 1983, which had left him partially paralysed. He used a wheelchair and was in constant pain. He died on Christmas Day 2009, having taken an overdose of muscle relaxants. "For me it was like a day you always knew was going to happen," Wagner says. "If you were friends with Vic you sort of accepted that that was a possibility. It was just part of the package of being his friend – a strange part of it." Wagner says Chesnutt wasn't good at being himself, that he'd tried to kill himself several times, and "sooner or later he was probably going to be successful". The pair would talk through Chesnutt's despair, arguing about someone's right or otherwise to do whatever they wanted with their life, even if that meant taking it. "There was never a conclusive result of these discussions – there were these two different opinions about things, understanding each other's point of view. But in the end Vic was very persistent about it."

Wagner evidently finds it difficult to talk about his friend. He's not big on eye contact at the best of times, but while he remembers Chesnutt his eyes are downcast, and his still hands begin to pick and pull at the seam of his trouser leg. He says he's talking about it more than he should. He's worried he'll be accused of exploiting Chesnutt's memory. He's perhaps more worried by the unjustified fear, lurking at the back of his mind, that he actually is exploiting Chesnutt's memory. "After he passed away, there was some sort of a tribute or a benefit that happened in Athens [Chesnutt's hometown] and we all went down there, and it really left a strange taste in my mouth," he says. "I wondered what I was doing there. It just became more of a platform for bands – a promotional situation – and it really grossed me out, and I realised that what I felt was much more private and personal to the point where I didn't really feel I should share anything about it, except with other friends that were close to him."

In truth, without the dedication, the casual listener would be unlikely to work out that Mr M is an album of mourning – Wagner's writing style is allusive and impressionistic – but he feels he has opened the door for those who might be suspicious of his motives: "I did it, and now I have to answer for it."

He has no time, though, for the idea that confronting his feelings through his work could offer any comfort through the grieving process. "The idea that music is some sort of therapy or art is some sort of therapy, I've never really bought into myself," he says. "I imagine it does help some people. But the idea that it's therapy is an admission there's something wrong with you and it needs fixing." Music is his job, he points out firmly but politely. "It's just what I do. If I was a mechanic I would have gone on fixing cars."

It's more likely, he says, that people find consolation in the music of others, often in the most unexpected way. "Vic's got a funny story about how actually music did help him through once," he says, lapsing absently into the present tense. "We were gonna do a US tour with him and the day before we were gonna leave he had taken off in the van and just disappeared. He turned up in Florida in some crappy hotel. He'd filled his pockets full of rocks and all these heavy things and was sitting there by the pool trying to decide whether to throw himself in and sink to the bottom. Then this Cyndi Lauper song came on the radio and there was something in the song that made him stop and not do what he was planning on doing. It was True Colors or something like that. And I was like: 'You're kidding?' And he goes: 'Really, it was something as dumb as that.' But it gave him enough pause that it actually made him stop. And next thing you know he's surfaced. He's called, he's gonna be all right."

Mortality has been in Wagner's thoughts in recent years. As well as Chesnutt's death, he had his own scare – his "cancer thing" he calls it – when he was diagnosed with prostate cancer a few years back. Those worries have faded, but he's now become someone who notices the ages of people who have died. "I talked to my dad about it. He was like: 'I read the obituaries now, to see if my friends are in.' And I'm like: "Yeah, I'm starting to do that, too.' That's the whole thing with living. You just don't think about this stuff when you're younger, and why should you?"

When he was being treated, Wagner came face to face with the cost of healthcare in the US. He already pays $12,000 (£7,500) a year for his health insurance – "I pay out the ass for a minimum of coverage. I certainly can't afford what I pay" – and suddenly was lumbered with bills for a further $80,000. "It's an illusion to think that even when you have health insurance that you're not going to become financially distraught by having something happen to you. The truth is that it could have been $250,000 in debt. Is that satisfactory? No, it's not. That issue came up with Vic, too. Vic had health insurance but was not sufficiently covered – like any of the rest of us." He's furious that the richest nation on earth can't offer its citizens healthcare, can't guarantee them jobs or homes. He wishes Barack Obama wouldn't be so apologetic in his efforts to change things: "To me, he's still trying to be a little bit too polite in trying to hold out for some sort of reasonable discussion. I'd like to see him point some fingers at the people who are causing the problems."

Lambchop nowadays is a much smaller beast than when Nixon propelled them to a measurable level of actual success at the turn of the century, when band members numbered somewhere in the teens. "Right now the core runs to five or six people, and a lot of it has to do with economics." Put simply, Wagner can't afford to maintain a huge band, which is why he's always turned down offers to perform a Don't Look Back-style recreation of Nixon on stage. "I don't know how we did it the first time. Now things are a little more expensive and it's impossible. How can I compensate the band for all that work? 'Here's 200 dollars'? I don't think people understand it just trickles down and in the end there's not much left for the next man in line." Well, there's that, and there's his fear of marking Lambchop down as a group living in the past: "It puts you in a position as an artist that you're signalling better days are not ahead any more. It's the oldies circuit."

How long Lambchop can continue to make music people will want to hear is preying upon Wagner's mind. "Realistically in five years I will be 60, or close to it. Is that something someone's gonna want to see or care about?" he wonders. But it's now 12 years since he gave up laying floors to become a full-time musician, "and now I'm starting to wonder what else I'm fit for at this point in my life, and I honestly can't think of much else."

Mr M is released on City Slang on 20 February. Lambchop tour the UK from 1-7 March. lambchop.net