Delaware and New Jersey are not reciprocal states, i.e., there is no special treatment to make it easier to do your taxes with both.
So, if you haven't filed yet, I would do Delaware fist as a non-resident, then I would do New Jersey as a full-year resident. Residency is a function of the facts on the ground, not what state happened to get your withheld taxes. Besides, it happens all the time that you live in one state but have to report income in another.
If you are asked, allocate all earnings to New Jersey. Besides, the New Jersey return will probably be aware of your Delaware wages - many states take the entire annual income and allocate it on residency time or whatever...
