the w-2 should be corrected to report the wages paid to you for the 2-3 weeks you lived in the first state but no change in withholding
and the wages paid to you after you moved to the other state - no withholding since none was taken
while it may not be able to change the withholding, it should be able to issue a w-2 correcting the wages.
you can not change the withholding.
the proper thing it would seem is to file a part-year return for the first state (reporting the wages paid, if any, for any work performed there). it would be an amended return if you have filed an original return. there is no way to know if the state will accept the return without the corrected w-2
if the state wants a corrected W-2 but the company won't issue it, it would seem your only recourse would be a lawyer which might cost you more than what your refund would be
