If you have not changes your residency to NY then you will file a non resident NY return and a resident MA return but get a credit for the double taxing of the income .. complete the NY return first and then the MA.

Now if you change your residency ( DL, voters registration, ect) then you will file part year returns for each state.

Thank you for your response. How does the NY credit work? Say I my income is 10k, paid 6.33% NY state tax->$633. I would owe 6.25% to MA tax ->$625. In this case, I don’t have to pay any MA state tax since 633 is enough to cover 625?

The "other state credit" comes into play only when you live in one state and work in another, and thus have income taxable by both states.  If you stopped working in MA before you moved to NY and became an NY resident, then you'll file as a part-year resident in each of the two states* - and there won't be any credit.  Your income as an MA resident will be taxable by MA; your income as an NY resident will be taxable by NY.

 

*Note: you will only have to file a part-year resident MA return if your MA gross income exceeds the filing threshold of $8,000.   https://www.mass.gov/guides/personal-income-tax-for-residents#-filing-requirements-

 

Assuming your move was permanent, you became an NY resident on the day you began living in your new home in NY.   That is what determines your residency for state tax purposes, not the date of your new drivers license or voter registration.