New York's instructions are to use Form IT-203-X if you mistakenly filed Form IT-201, but you were a nonresident or part-year resident.
https://www.tax.ny.gov/pdf/current_forms/it/it203x_fill_in.pdf
When you do the amended return in TT, be sure your residency information is correct in the personal info section. Your "state of residence" should be NJ (the state of which you were a resident on 12/31/2020), and be sure you indicated that you lived in another state (NY) during that year. The move date is the date you began living in your new permanent home in NJ. This should prompt the program to generate Form IT-203-X. (I'm not 100% certain because I don't have the NY software on my computer.)
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
