Adult male peeing when he goes into a room without us, or I leave him alone?

The fact that your dog does not pee in a crate is to be expected. That’s one of the major benefits of crate training. Dogs do not want to soil where they’re going to sleep, so unless the dog really can’t hold it they generally won’t go and if crated might cleverly angle to eliminate just outside their crate. This dog may feel most at comfort while in his crate alone. So it may be a good idea to continue crating him while you’re at work

Since you just rescued him a couple weeks ago, I would suggest your senior (9 yrs.) dog is going through a Lot of Life Changes- his previous owner/family/home is gone, now he’s at a new place, with different humans, and other dogs, who he doesn't sound close to yet. In summary, it sounds like nerves are due to his peeing in the home. What you’re describing, (full bladder dump/emptying) differs greatly from marking (when a small stream is deposited, usually on furnishings to mark).

Until this dog acclimated and really feels at home, you likely have to do more: When you get up and go to another room, take him with you, even if you plan to go out, that may require placing a leash on him. Or crate him.