Since you actually worked only in NY, per the NJ reference cited by @SweetieJean , your 2021 income earned prior to October 1st would be taxable by both NJ (the "employer's jurisdiction") and NY, and your income from October 1st to the end of the year would by taxable only by NY.
On your NY return, you would be able to claim an "other state credit" for the taxes paid to NJ, so you wouldn't be double-taxed.
This TurboTax reference explains how to amend a state return only:
I need to amend my state return (intuit.com)
Be sure to complete the non-resident NJ return before the home state NY return, so that the program can calculate and apply the credit correctly.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
