You can only be a resident of one state at one time. So you could file Part-Year Resident forms for both Delaware and New Jersey, or you can file one as a Resident and one as a Non-Resident. The wages on your Form W-2 will be taxed where the work is performed, so that sounds like New Jersey but it could be on a Non-Resident or a Resident form.
Residency of a state also takes into account intention. If you kept your Delaware driver's license and were still registered to vote there but were temporarily in New Jersey while you worked, you might make a case that you were a resident of Delaware all year and a non-resident working in temporarily in New Jersey.
If you moved with the intention of taking up residence in New Jersey, changed your license, voted there, and changed all your mailing addresses to New Jersey, a case could be made that you did become a resident of New Jersey, and you could file a part year resident return for both states.
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