@em460
When TurboTax asks for your mailing address, enter your current address. That's the address that goes on your tax return.
The address on your W-2 doesn't matter, but on the screen where you enter your W-2 in TurboTax there are boxes for the employee's address. TurboTax will automatically enter your current mailing address. Change it to the address that's on your W-2 form, even if it's wrong or you don't live there anymore.
If you moved in 2019 you will have to file part-year resident tax returns for both New York and New Jersey. When TurboTax asks for your current state of residence select New Jersey and click Yes where it asks if you lived in another state in 2019. Select New York as your previous state, and enter the date that you moved from NY to NJ.
You didn't say which state you worked in. If you worked in NJ while you were living in NY you will also have to file a part-year nonresident tax return for New Jersey.
