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.