I just adopted a 1 year old rescued Shih Tzu and he seems to be very shy. He's not moving his hind legs or drinking/eating anything. I haven't even heard him bark. Is this normal behavior when dogs are first being brought into a new home?

Inability to move move vs. reluctance to move?
No this does not sound normal, even for a timid dog.
Time to get him checked out by the vet to determine if his problems are medical.
The phrase that concerns me here is this " He's not moving his hind legs" what about the front legs?
A painful injury, will affect their appetite, not saying that is what is going on simply that your word choice makes me think something besides new dog may be going on here.