TT flags this as a problem. every state that I've ever worked with that had an income tax required an ID # so it could match the withholding you claimed with what the employer showed. it could be that the copy it filed with the state had an ID # in which case getting it from the employer would be adequate without it supplying a corrected w-2. if the state's copy doesn't have an ID #. likely the state will contact the employer but no telling when that would be - inform your employer to get things moving.
