From your statement, it would appear that you are a part-year resident of MA and a part-year resident of NY, and would file each state return as such. I have listed below links that should assist you with each state's filing requirements. You say you had MA tax withheld for the year so what you will do eventually with filing is receive a refund for the period of non-residency with MA and, unfortunately, turn around and pay it to NY for the NY period of residency.
The fact that you still have your voter registration and drivers license in and for MA does not matter.
Since the periods of residency in each state do not overlap, there would not be a credit for taxes paid to another state taken on either return. As you are now a NY resident, when filling out your state information, enter your data for MA first, as it is now the non-resident state, and then fill in your data for NY. That is the procedure for entering multiple state data in TurboTax. Non-Resident state first, then resident state.
MA Personal Income Tax for Part-Year Residents
NY Part-Year Residents
@lp601
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