Lived part time in NJ and NY

Yes.  You'll file part-year resident returns in both states and the nonresident periods will be handled within those returns.  

In the "My Info" section, edit your personal information and indicate which state you lived in on 31 Dec 2023.  Then answer, "Yes" you lived in another state and select the prior state from the list.  Enter the date you moved to your new state.

After entering your income and deductions in the federal interviews you can start the state returns.  You will allocate your income in each state's interview based on where you resided when you received the income.  

It is strange that zero was withheld for NJ so you likely will owe taxes there.  Be sure to check with your current employer to get withholding started for the correct state going forward.

You said the w2 shows no NJ income but that doesn't convince me that is the case. Not knowing about your income, you may need to file a NJ NR form in addition to the resident form. See page 4 of 
Nonresident Income Tax Return Instructions for your situation. Whichever state has the income and creates the tax liability should be done first and then the state that will credit the amount should be entered.