It depends. If you earned money while living in NY, then yes, you will need to file a NY Part Year resident return. If you did not earn any money while living there, you would not need to file a NY return.
For states, you must file a return in your home state if you earn money from anywhere in the world or if it is not your resident state, if you earn money in that state. So if you are living in VT and working in NY, you would have to include the income in both NY and VT.
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