Since NJ and NY have different definitions of a resident for income tax purposes, it is possible to be considered a resident of both states. In the situation that you describe, that's exactly what would happen. New Jersey is your domicile, which makes you a NJ resident. An apartment in NY is a permanent place of abode, even though it's not your primary home. So if you are in NY for 184 days or more, you are a NY resident. See the definitions of "Permanent place of abode" and "Resident" at the following link.
NY Income tax definitions
Being a resident of both states would mean that you would have to pay income tax to both states on all of your income. I don't know if you would be eligible for the credit from either state for the tax that you pay to the other state, or how the credit might be calculated.
This is an unusual and obviously messy situation. I suggest that you consult a local tax professional, or a tax lawyer, for advice. Depending on what the tax pro or lawyer tells you, you might want to consider changing your plans in order to avoid the situation.
