Q. Do I file taxes for both states?
A. No.
VA & WV have a reciprocal agreement, on wages. No VA taxes are withheld or due and you do not normally need to file a VA return. But WV will tax you on your other state income.
If VA taxes are mistakenly withheld, you then have to file a VA return, to get a refund. WV will not give you a credit, for taxes mistakenly withheld by a reciprocal state.