David Nail Is a 'Worrier' Kind of Dad While Wife Catherine Is 'Kind of a Hippie'

David Nail‘s twins will be turning 1 next month, and the new dad is still finding out just how much of a worry-wart he is when it comes to his kids.

“My wife’s kind of a hippie here . . . not much of a worrier,” the singer, 37, told PEOPLE on the red carpet at the CMA Awards on Wednesday. “I’m the dad that I have to see them at all times. If they even remotely look at the drawer that they could potentially open, I have to run over there and pick them up.”

And though he thought he was ready for fatherhood, “you just find out so many things about yourself that you thought you kind of had an idea of but in reality have no clue,” he says.

Becoming a dad to his son Lawson Brent and daughter Lillian Catherine (with his wife Catherine) has also made him reflect on how important his loved ones are.

“People used to tell me all the time, ‘Hey, when you actually have kids, you’re gonna realize how truly significant family is,’ and I think . . . you get that it makes sense, but there’s nothing that can prepare you for when you see your kids,” he says.

Adds the star: “You automatically think about how much your parents loved you and just how proud they are of you and how much they cared for you.”

• Watch PEOPLE and EW’s red carpet LIVE show, airing Wednesday, Nov. 2 at 5:30 p.m. ET on our homepages or the new People/Entertainment Weekly Network (PEN). Go to PEOPLE.com/PEN, or download the PEN app on Apple TV, Roku Players, Amazon Fire TV, Xumo, Chromecast, iOS and Android devices.

People Special Edition 50 years of the CMA Awards – Cover

• PEOPLE’s Special Edition 50 Years of the CMA Awards is on sale now online and on newsstands.

Nail also joked he’s occasionally felt like a “crappy dad” because nothing “changed overnight” musically for him. However, he now sees his past work in a different light.

“The one thing that I did realize was not so much that songs that I was writing or that I felt suddenly changed at the time,” he says. “It was songs that I’ve already recorded, songs that were on my record already that suddenly there were lines in there that maybe caused a few more goosebumps than they did previously.”

Adds the singer: “There’s no guarantee my kids will ever think anything I’ve done is cool, [but] there’s a few songs I’ve written about their mother or about them … they’ll appreciate from the standpoint that it came from the heart.”