It depends upon how you were brought up, and what you're familiar with. I went to a parochial high-school in the late 60s, and coming out of the showers everyone was in towels, and changed back into underwear the way you described. In the Army in the 70s, the showers were about equally divided between guys who covered up and changed under towels, and those who just were naked till they got dressed. Years later, for several years I patronized a gym on a military base. At first, I covered up, like I was used to. But the military or retired military guys who used that gym never covered up at all, everyone walked nude through the locker-room, and never gave it a second thought. So pretty soon, I did, too. I always changed facing the wall, though, maybe some residual modesty.
Now I go swimming in the winter at a local high-school, the swimming-pool is open to the public on Sunday afternoons, and use the showers and locker-room there. It's used by males of all ages, including little boys, and I've gone back to covering up. Last year, an incident happened that was kind of humorous. I'd just gotten out of the natatorium (swimming-pool area), and opened the door to step into the locker-room. A small boy was next to the bench changing into his clothes, but he was naked, and had evidently just dropped his towel down. He got an awfully embarrassed look on his face, said "I'm sorry", and started to apologize. I answered, "Don't worry, this is a locker room, it's okay".