The "other state credit" is granted by the state of residence. NY will credit you for the taxes paid to NJ during the portion of the year that you were an NY resident. NJ will credit you for the taxes paid to NY during the portion of the year that you were an NJ resident.
After your move to NJ, your income will continue to be taxable by NY (as well as by NJ) if you worked remotely from the NJ location for your own convenience, rather than because you were assigned to or required to work in NJ by your employer. This is known as the "Convenience of the Employer" rule. Details here:
https://www.tax.ny.gov/pdf/memos/income/m06_5i.pdf
NJ's tax rules for non-residents and part-year residents are detailed in this NJ tax publication:
https://www.nj.gov/treasury/taxation/pdf/pubs/tgi-ee/git6.pdf
Here are TT's instructions on filing part-year resident state returns:
How do I file a part-year state return? (intuit.com)
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
