No, you should not adjust the values shown on your W-2 when you are completing your return to account for the dates of your move. You're W-2 should be entered exactly as you have received it.
However, when you are working through the two part-year resident returns you will be able to allocate the correct amount of income to each of the states so that it will not be double-taxed. Since one state had more tax withheld that you expected, that should mean that you will just have more of a refund or less balance due for that state.
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