Rent move from NYC to NJ

Hello:

In 2020 I moved from NYC to New Jersey in February, and i received W2 with all tax to NY.

when filing tax form, I was told to file part-resident to both NYC and NJ, and I have very little refund from NY state, and have to pay full NJ tax.

So I selected "I double paid both state tax" and then the tax due becomes about 12 USD in NJ state. After doing that NJ + NY tax looks more reasonable comparing to my 2019 tax return. Because my salary and employer didn't change.

Question is : Do I need to select that "double tax payment" ? In W2 I paid almost same amount as 2019 but I only lived in NY for less than 2 months, and it barely return any tax, if I don't select that I'll have to pay another full amount NJ tax.

Thanks a lot