You have to file a part-year resident tax return for the state that you moved out of. You do not have to say anything about your W-2. The W-2 correctly shows the tax that was withheld. The employer sent that tax to the state for which it was withheld. There is nothing that needs to be updated on the W-2. It doesn't matter if the tax was withheld for that state was more than it should have been. The W-2 shows the amount that was actually withheld. You get the excess back by filing the part-year resident tax return, not by changing the W-2.
If you have further questions, please tell us what states you lived in, what states you worked in, and what states tax was withheld for.
