You file as a part-year resident of each state. This previous TT answer explains how to do that, including the recommended sequence: https://ttlc.intuit.com/community/state-taxes/help/how-do-i-file-a-part-year-state-return/00/26057
This previous TT answer has some info on doing the part-year NJ return. (Change the year in part 5 of the answer to 2019): https://ttlc.intuit.com/community/state-taxes/discussion/can-i-file-both-a-resident-and-non-resident...
Your "double-taxed income while a resident of NJ" is the portion of your wage income received after you became a resident of NJ. You became an NJ resident on the day you began living in your new home in NJ.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
