We got a shelter dog a few years back and it took us quite a while to house break her. Dogs usually enter the shelter because
they're not house broken and the original owner hasn't done anything to teach them. Then they end up in the shelter where those poor things aren't walked that much and they learn to pee/defecate in their cages.
As ehbeth said, after only one week you cannot expect the dog to adapt
that fast. You have to be patient and re-train the dog anew. Go out every hour and just let him pee, the minute he did his business, be overly excited and tell him he's a good boy. Do walk him 3 times a day for long walks, but the hourly trip should be just as long as he pees.
Be patient with the little guy, he's so eager to please you, but just doesn't know how to yet. It's not easy to adjust to a new home, especially when there are 4 year old twins running around too.
I am so glad though that you took a shelter dog - you'll see, there
are the best and most grateful dogs to have. Good luck!!