By moving out of NYC to NJ, you don't pay any more NYC taxes. You work for a New York company, and New York taxes is withheld from your paychecks. Besides NY part-year return, you also file NJ-1040 resident return, and make sure to claim the credit for taxes paid to NY by filing Schedule NJ-COJ Credit for Income or Wage Taxes Paid to Other Jurisdiction. You may also qualify for renter's credit if you lived in NJ for over 6 months. If you paid enough taxes if NY and had no NJ source of income, you may receive a small refund (around $50 renter credit).
TurboTax can handle your tax situation.
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