Since you moved during the year, you will file a Part-Year return in New Jersey (which is technically a resident and a nonresident return) and a Part Year return for New York. On the New Jersey return (complete New Jersey first), you will allocate the income as resident and nonresident.
Then you will do the Part-Year New York return to get the credit for taxes paid to New Jersey- just make sure you have the nonresident NJ income included. And of course, you must pay New York City tax for the time you were a resident. With the credit, they amount you owe to New York state should be pretty close to your credit but the NYC tax adds a bit if you did not have NYC withholding.
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