You file your 2021 tax returns, and pay the tax, exactly the same way as if you didn't move. What you do in 2022 has no effect on your 2021 tax returns. The only thing you do differently is to enter your new address when TurboTax asks for your mailing address, so that any mail from the IRS or your old state will go to the new address.

And....you make sure that your employers know your new address so that if they are mailing your W-2's they have the correct address to mail them to.