In a cramped rehearsal room on a side street in Brooklyn, AJ Lambert and her band are quibbling over the order of their set, trying to make sure it contains contrast and dynamism. It includes a handful of covers of songs by Shame, Spoon, TV on the Radio, Robert Wyatt and Chris Bell, all radically repurposed by the band of two synths, bass and drums. The setlist also contains a handful of Sinatra standards and these need to be especially well-judged. After all, Sinatra was Lambert’s grandfather.

“When I hear things he sings, I hear them through some other filter,” she tells me later. “I hear them as a fan, but also as a human being I knew.”

The placement of one song they are particularly puzzling over is that of Glad I’m Not a Kennedy, by Shona Laing. It’s a startling song to hear in this context, because of Sinatra’s history with the Kennedys. He was John F Kennedy’s friend and cheerleader until being abruptly cut off by the family in 1962, for reasons that have been the source of endless speculation since: usually assumed to involve the mafia, sex or both.

“Yeah, that song’s super-personal to me,” Lambert says after rehearsal, eating a slice of salted caramel apple pie in a cafe around the corner. She knows how deeply her grandfather was hurt by his excision from Camelot’s inner circle. She knows, too, how he grieved after Kennedy’s death. But she has no idea what happened, either. “A lot of times I don’t know if I get the real story. Because they’re” – they being the Sinatra organisation: the family, the lawyers, the management – “so used to being on the defensive, understandably. But I’m far enough removed from all of it to be: ‘Well, I don’t know what’s true.’”

AJ Lambert: Careful You – video

So when Lambert sings Glad I’m Not a Kennedy, it’s not so much with the original’s regret for political opportunities missed, more with a sense of sadness for people whose lives have been overshadowed by their name, something she understands a little. “I’m singing it because they say all this horrible shit about my family, but it must have been so much worse to be in that family with all the shit that gets said about them. It’s a sad song. There are so many levels of tragedy and death and misery in that family.”

Lambert is 44, and only now releasing her debut album, the excellent Careful You, in which the indie tracks and the standards come together in a seamless whole that recalls the lush soul productions of Matthew E White’s Spacebomb stable, topped by Lambert’s restrained, insinuating voice.

She has been delayed by alcoholism (she’s four years sober), by motherhood, by having other jobs. And by not wanting to be seen to be hanging on the coattails of her grandfather or her mother, Nancy Sinatra. She was in indie bands through much of her 20s and 30s: a group called Sleepington at college, one called Looker, which ascended to the dizzy heights of recording a session for Steve Lamacq, brief spells with Here We Go Magic and the reformed punk band the Homosexuals – but tucked away in the rhythm section rather than out front.

“I loved it,” she says. “I was a wild kid, drinking and drugging, and I loved it. But it was a world that suited me at that time. Perfect life, perfect fun. Until it wasn’t.” Her alcoholism, she says, was insidious. “I would black out pretty much every day. I don’t remember chunks of my life and nobody knew. It was that scary thing; I could be sitting here having this conversation looking normal. The next day they’d tell me we had a great conversation, and I wouldn’t remember it. All the time.”

The singing only began in earnest in 2015, with her and a pianist doing covers of her favourite indie songs in a lounge act (“but it wasn’t campy; it wasn’t ironic”). In March 2016 her uncle, Frank Sinatra Jr, died, “and he left this hole, where no one in my family does that music at all. My mom doesn’t do it. And they approached me, the family powers that be, management types. ‘OK, we know you’re singing. He had been about to do this tribute show: do you want to see the list of songs?’ It was every single hit there ever was. And I felt if I were to do that it would be so cynical. It seemed ridiculous to be a 41-year-old woman, at that time, singing Love and Marriage and My Way. I said: ‘I don’t feel comfortable doing this, but I’m so honoured to have that lineage, and I feel like I have to do something for it.’ So I decided to do [Sinatra] album shows, where I do In the Wee Small Hours and Only the Lonely. This record came from having that experience.”

AJ Lambert (second right) with Frank Sinatra, Amanda Lambert, and Tina Sinatra Photograph: Alan Berliner/BEI/REX/Shutterstock

Being the child of showbiz royalty wasn’t quite as bonkers as one might expect; visiting her grandfather was not a journey into the Rat Pack. “Everyone wants it to be jetting around and singing My Way around the piano every day. It just wasn’t like that.” There was wealth – of course there was – but not unimaginably so. Sinatra never wrote a song, so never saw a penny in publishing. Lambert gets a fixed income from the family’s ownership of the Sinatra master recordings and the name and likeness rights (“But it’s not enough to live off, because there’s a lot of us”).

Relatively normal, though, does include the wholly unusual, such as when Nancy, then 54, posed naked for Playboy in 1995. “Can you imagine your mom doing that? I was miserable about it. I made a documentary short film about it, because I couldn’t process it. I made the choice: ‘I am going to be OK with this for now, because it’s the only way. I can’t sit and say it sucks, because it will make her feel bad and me feel worse.’ I thought, she must have a reason to do this. And she got paid a lot of money. Maybe she needed the money.”

But everywhere, at all times, there’s the knowledge that everyone who meets her is thinking: that’s Frank Sinatra’s granddaughter. And the peculiarity of the fact that the person who to her was a grandad is to everyone else – literally, everyone else – less a person than a representation of something mythical, despicable or captivating, whatever it might be, but always just a representation. “That’s the thing,” she says. “He’ll always exist. It’s almost like they’re two different people for me. I can see him as the totem and I can see him as the person, and they’re very different people, but I know both of them.”

And when you hear her sing the songs he sang, there’s a moment when you can understand that.

• Careful You is released on Alpha Pup on 25 January. AJ Lambert plays the Moth Club, London, on 24 January.