How do I know if the state taxes were paid when I filed my return?

How you chose to pay the state taxes owed will be on the State Filing Instructions page.  Use Print Center.

To access your current year online tax returns sign onto the TurboTax website with the User ID you used to create the account - https://myturbotax.intuit.com/

When you sign onto your online account and land on the Tax Home web page, scroll down and click on Add a state. 

This will take you back to the 2021 online tax return.

Click on Tax Tools on the left side of the online program screen.  Then click on Print Center.  Then click on Print, save or preview this year's return.  Choose the option Include government and TurboTax worksheets

What arrangement did you make to pay your state tax due?   Only your own bank or credit card record will show that you paid the state.