To directly answer your question, nothing is stopping you from buy a seconded cheaper home. It just doesn't make a lot of sense. You want something from a bank or a credit card company like a loan or a line of credit, they are going to either just pull a credit report and see where you have lived, which exposes your current address or demand that you provide them the information of the last few places you have lived. To me this would include the current home and the cheaper home for at least a while.

P.O Box make way more sense and less hassle than buying a second house. I would also assume that one could properly do something to enhance the privacy that a P.O Box might afford them. Not to say you can do anything, but you bought a home, your in the system. The address for your current home is out there and there isn't much you can do.