You'll have to file a part-year resident return in each of the two states (assuming each state has an income tax). This FAQ explains how to do that in TurboTax: https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return
You'll receive a refund of the excess withholding from your "old" state, but you'll have to pay your taxes due to the new state.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
