The IRS does not care if you live in NJ or NH--the IRS is not concerned with state income tax. The IRS is concerned with your federal tax--the tax paid to the U.S. government. Nor does the IRS have any interest in your drivers license.
You do not fill out a W-2 for an employer--the employer issues a W-2 to you in January. What you fill out for an employer is a W-4. The W-4 indicates to the employer how much federal tax to withhold from your paycheck. It does not go to the IRS.
New Hampshire does not have a state income tax, which is why no state tax is being withheld from your paychecks now that you live in NH.
https://ttlc.intuit.com/community/state-taxes/help/which-states-don-t-have-income-tax/00/26063
If you moved to NH during 2019, you will still have to file a part-year NJ state tax return when you prepare your 2019 tax return--in early 2020. You will be able to indicate in the software that you only lived in NJ for part of 2019. In January 2020, you should receive a W-2 from your NJ employer and your NH employer (if you have a different employer now) Your W-2 will indicate how much NJ tax was withheld from your pay while you lived in NJ.
As for the drivers license--check into the laws in NH so you know how long you have to obtain a NH drivers license in order to comply with the state's laws regarding drivers licenses and auto registration.
**Disclaimer: Every effort has been made to offer the most correct information possible. The poster disclaims any legal responsibility for the accuracy of the information that is contained in this post.**
