I have switched my residence from NJ to NY in 2019 and worked and lived in New York for the entirety of 2020.
However, my W2 form shows both NJ and NY. Do I need to get a new W2 form to correct this? I was wondering if I can just use my current W2 form to file my Federal and New York tax.
Right. You should ask your employer to immediately stop withholding taxes for NJ since you don't have a tax liability there.
That is, New Jersey didn't do the withholding, your employer did. Ask your employer to stop, since you don't have a good way to get this particular tax back.
You will want to file a Non-Resident NJ tax return to get your NJ withholdings refunded.
Even though I have changed residency during 2019, I have worked in NY all of 2019 and 2020. Seems like only thing withheld by NJ was SUI/SDI, which I cannot get back by filing non-resident tax. So I don't need updated W2 form right?
Right. You should ask your employer to immediately stop withholding taxes for NJ since you don't have a tax liability there.
That is, New Jersey didn't do the withholding, your employer did. Ask your employer to stop, since you don't have a good way to get this particular tax back.