Need help with my dog!

A few months ago we adopted a six years old dog. He's wonderful but there's one issue and it's kind if big. He pees inside. The shelter neutered him, so he had only been neutered for a short while when we got him. We go on regular walks. During the day it's usually no problem but at night he'll go around and mark his territory. I learned quickly that I can't have new stuff on the floor because he'll pee on them, but he'll also randomly pee on "old" stuff. I started get up at night to get him outside, but that doesn't help. I can't crate him because it's illegal where I live. The only thing that seems to help is to keep him locked in the bedroom with me at night, but I have a three years old son who usually sleeps in my room and gets up at night to pee and go to his room and back so we both forget to close the door very often, and every time it happens I wake up to a pee clean-up party. It's nasty and I'm sick of it and I don't know what to do. We've been at the vet and he's healthy. I spray anti pet pee spray whatever it's called but it doesn't help. He gets enough exercise. I have put up pet gates throughout the house to limit access. I rarely catch him in action so I can't scold him. I praise him when he pees outside. I'm at loss on how to stop the peeing.