Your rights depend in part on whether you have a lease. If you have a lease, your lease would remain in place, except some circumstances. If you do not have a lease, the new owner would be entitled to evict you. They would have to give you 60 days notice, and maybe 90 if you qualify under federal law. However, if you are in default, don't pay, then the rules of eviction based on default would apply. So you should pay your rent to the current owner, and keep records of payment. It may be the owner is trying to do a loan mod to stop foreclosure and is relying on the rental income to do so. If the landlord can't make the loan mod. payments,then you both lose.
If you are worried about moving money, new owner often offer "cash for keys". This means they offer moving money in exchange for agreement to leave so they same time and avoid eviction lawsuit.