"nor do i have any specific state sourced income"
If your domicile is now in a foreign country, then in the Personal Info section of TT you should enter your State of Residence as "Foreign or U.S. Possession" - which is at the bottom of the State of Residence drop-down list.
Since you moved from NJ during the tax year, you should indicate that you previously lived there. The date of your move should be the date you began living in the foreign country.
If you had no state-sourced income after your move, then indicate that you had no "Other State Income".
Making these choices will prompt the program to generate a part-year resident NJ state return.
Your mailing address should be the address you want the IRS or state taxing agency to use in case they need to contact you about your return. If that happens, they will contact you by physical letter.
You can use a foreign mailing address on your federal return. See Foreign Address on page 14 of the Form 1040 instructions for details on how to do this:
https://www.irs.gov/pub/irs-pdf/i1040gi.pdf
It is also possible to e-file with a foreign address using TT. See this:
https://ttlc.intuit.com/turbotax-support/en-us/help-article/electronic-filing/e-file-foreign-address...
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
