With a lot of boy dogs peeing is part of claiming their territory, is your other dog a male too, that could be the problem, I think he is sooo happy to have a good home at last he is staking claim to it. You will have to be quite severe with him if you are going to get on top of this problem, when he pees tell him he is a "BAD BOY" in a very loud voice then take him outside, when he does wee outside tell him he is a "Good Boy" and give him a treat.
If all else fails he could be like my old Springer Spaniel Ben, if you can't eat it or make love to it, pee on it, he eventually grew out of it, he would then only pee on judges at the dog shows.