We adopted an adult dog too, and at that time he was not housebroken. When we brought him home he immediately peed on the wall.

But that was the only time and after that he did not pee in the house anymore. We had more problem with pooing in the house, but since both of us worked he was left alone for about 7-8 hours right after we got him. For about 3 weeks we have restricted him to the living room when we were not home and covered the room with newspaper. He was completely housebroken in less then a month. We rewarded him every time we got home and he had no accidents. Never punished him for the accidents though, since he was from a shelter, so we did not want to cause more trauma to him. Instead all we had to do is change the tone of our voice and he got the message.I think dogs when they mark - peeing everywhere - they usually pee where other dogs have peed before them. And being neutered or not does not make much difference, i had them both. The initial hardship, and there is going to be some, well worth it, they will appreciate you for life.