Yes you will probably need to file in both New Jersey and Delaware.
Part-year Delaware residents must file if they received income from any source while a Delaware resident. (so you would need to file unless you received no retirement income in December)
Part-year New Jersey residents whose gross income from all sources for the entire year is greater than $20,000 ($10,000 if single or filing separately) must file Form NJ-1040 to report income received while a resident and Form NJ-1040NR if you have any income from a New Jersey source while a nonresident.
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