State tax filing

NH doesn't have an income tax on wages, so you won't need to file a NH return.   For NJ:

The NJ Division considers many factors when determining if New Jersey is your domicile, including: 

•  Your intent; 
•  Where you register to vote; 
•  Your driver’s license and vehicle registration; 
•  Where you have family ties; 
•  Whether your federal tax return lists New Jersey as a home address; 
•  Location of bank accounts; 
•  Whether you participated in a New Jersey property tax relief program.

If you became a resident of New Jersey or you moved out of the state during the tax year, you are considered a part-year New Jersey resident.  Part-year residents may be required to file a New Jersey tax return.   Since New Jersey does not have a special form for part-year filers, you must use the regular resident return, Form NJ-1040.  The return provides a line for you to show the period of your residency in the state.

As a part-year resident, you report only the income you earned or received while you were a New Jersey resident. You must prorate all the credits, exclusions, exemptions, and deductions for which you qualify. That means you can take only that part of the full deduction, credit, exclusion, or exemption that represents the amount of time you were a resident of the state.

If you continued to receive income from New Jersey sources after you moved out of the state, or if you received income from New Jersey sources prior to moving to the state, then you also will have to file a New Jersey nonresident tax return, Form NJ-1040NR. In this case, you will file both a part-year resident return and a part-year nonresident return.

You must file a return and pay tax on the income you earned if (see chart linked here:(