Q. Should I pay state taxes ?
A. No.
Q. What should I do in this situation?
A. Nothing, unless there was state withholding (box 16 of form 1099-B).
The fact that your old address shows on the 1099-B (or any other tax form) is meaningless. Even if your old state postal abbreviation is in box 14 of the 1099-B (which is unlikely, you can still ignore it.