Most likely, the issue is that he has NOT been through a lot in his first two years, and every single thing is a new experience for him. Hoarders and puppy miller dogs are in such a fragile state. While I think doing some potty training is ok, that should not be your focus right now. Ozzie needs to see you as 'safe', his environment as 'safe', and these other dogs as 'safe'. Too much, too soon, will only overwhelm this poor overwhelmed pup.

Quietness, lots of walks, and let him get to experience you.

One thing I have heard is when he does piddle inside, take that outside, in the spot you want him to potty, and praise, praise, praise...though this might have to be soft whispers with this puppy.

Does he have a secure 'den' in your home? Away from others? I would think that would be a big deal for him right now.