As a boy, John Carter Cash used to be taken hunting by his father, the American music legend Johnny Cash. They would go into the woods around their home in Tennessee to shoot squirrels, rabbits and deer. Or they might go up to the wilderness of Alaska with his mother too, the country singer June Carter Cash, who was a skilful fisherwoman, and catch pike and trout. Along the way, Johnny Cash told his son all about nature – not just the beauty but the horror and bloodshed as well. Then they would drag their carcasses home, and skin or pluck them and eat them for dinner. "My father was a great outdoorsman," says Cash. "From when I was about six we would spend countless hours together in the woods or on a lake. He taught me how to skin a rabbit and pluck a wild turkey. He showed me there is much more to nature than we can ever understand."

As the 10th anniversary of Johnny Cash's death approaches, his son John, 43, has written a children's book – his first novel. Lupus Rex is dedicated to the memory of his father's brutally honest introduction to mother nature.

The book – part fantasy, part fable – is about a group of animals that inhabit a woodland kingdom. When their crow king dies, all the other crows gather to choose a new king. This murder – to use the collective noun for crows – turns against each other horribly, and what follows is a series of battles and betrayals as they fight to restore order in their land. Just like Watership Down, all Cash's animals are given brilliantly human characteristics. And as in Peter and the Wolf, each animal is represented by its own song in a gorgeous folky soundtrack that accompanies the book. The music was recorded by Cash, who is also a Grammy-award winning musician and producer, along with some of the Nashville greats, including Bill Miller, who worked with his father.

"The book is as much about understanding us as human beings as it is about animals," explains Cash. "I have one distinct memory from when I was a child of coming upon a group of crows and my dad saying to me, 'See those crows over there? See that one that's squawking? He's on trial and if he is found guilty he will be sentenced to death by the others. That's why they call it a murder of crows.' Well, that stuck with me since I was very small. I became fascinated with the idea of order and hierarchy in nature."

Cash's writing career didn't start until his early 30s, shortly after the death of his parents. They died within months of each other in 2003 after a legendary 35-year love affair and marriage celebrated in the Oscar-winning film Walk the Line.

"My mother's death was very painful as it occurred over a period of a week," says Cash. "Watching her die was the hardest thing my dad ever went through. Even though they had their struggles, they were built for each other, and by the end of their lives they were more in love than they had ever been. But at the same time he has a great sense of strength and endurance. On the way home from her funeral he turned to me and said 'I gotta get back into the studio. I've got to go face to face with this pain.'"

While his father spent his final few months in the studio, Cash began to write. "Shortly after Mum died, a publisher asked if I could write a foreword for a biography. When they read it they said, why don't you just go ahead and write the whole book? I was a little nervous because I knew it would be painful but I wanted to honour her legacy. So I took it on. In the end, it was cathartic."

In 2010, he found himself doing the same for his father with the publication of his biography The House of Cash. Again, he found the process of rifling through thousands of pictures and letters deeply healing. "It was more than just a reminder of exactly who he was," says Cash. "It made me feel like I was back in close contact with him again."

As the only offspring of the Carter Cash union, the responsibility for honouring their legacy seems to have fallen on Cash's shoulders. "I think of it as just one aspect of my life," he says. "If I didn't have all the other aspects of my life, my own writing and music, then maybe it would be overwhelming to me. But I could go on and on about my father and who he was. And my mother too – she is an important historical figure in her own right. She won two Grammys and she came from the Carter family, who basically made country music what it is today."

He may be country music royalty but life for Cash wasn't always easy. Born in Tennessee, where he still lives, he pretty much grew up on the road. His parents took him on stage at four weeks old and his father taught him some guitar chords just as soon as his hands could hold one. "I was getting up on stage and taking a bow as soon as I could stand," says Cash. "But they didn't force it on me. They weren't pushy. They let me choose my path."

But by his late teens, things started to go wrong. He began drinking heavily and was soon facing addiction problems just as his father had. "You would think growing up around addiction as I did, maybe I wouldn't go there but nobody is immune," says Cash.

He hit rock bottom at 21. "I went through hell," he says. "I did a lot of struggling with my identity trying to figure out who the heck I was. I had to face my demons."

He struggled until he was 30, when he finally managed to sort himself out. "I did a lot of growing up," he says. "I stopped banging my head against the same wall. I was frazzled but a lot stronger. Now I'm pretty comfortable in my own shoes. Now I'm OK with carrying the torch for the family heritage."

Which is lucky because it's a pretty major role. There's the Cash Cabin Studio to manage, where both his parents made much of their music. And this being an anniversary year, there is an album of unreleased music coming out, which Cash has helped produce, as well as a limited edition postage stamp featuring Johnny Cash's distinctive face.

"Sometimes I can't believe it's been 10 years," says Cash. "There are still days when I turn round to ask my dad something and then suddenly realise, oh, I can't."

Cash now has three children of his own and the legacy continues. Joseph, his eldest, works full time at the Johnny Cash museum, in Nashville. "He definitely gets a full dose of Johnny Cash on a regular basis," says Cash, "but I think he appreciates it."

Just as his father did with him years ago, Cash is immersing his offspring in the wonders of the outdoors. His daughter Anna Maybelle, 11, like her grandmother June, is a fisherwoman. "My little girl just loves to fish," he says.

But for now Cash is off. He is currently producing Loretta Lynn and is keen to get back to the studio, the cabin Johnny Cash built in the woods in 1978.

"We've had a lot of folks visit here," says Cash. "But it is the spirit of my dad which lingers on foremost. There's a photograph of him, the last portrait ever taken of him, hanging up in the main tracking room. He is watching over all the music being recorded here still. I can definitely feel that."

• The soundtrack of Lupus Rex is available from lupusrex.com