If you move during the year, you will file as a part-year resident of New York on form IT-203 -the same form that you files as a non-resident. Your NY income is taxed a little bit differently if some of it is earned while you are a resident and while you're not, which affects the credit on the New Jersey return. However, the New Jersey income is also adjusted for the period of residency as well, so it works out.
You would also still file the same form in New Jersey, NJ-1040.
As you might imagine, this type of scenario is relatively common!
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