How do I confirm that Turbo Tax paid my state tax so I do not owe anything?

TurboTax NEVER pays your federal or state tax due.  The only thing TT can do is pass your banking or credit card information along to the IRS or state so that the taxing authorities can debit your account.   Your only proof that you paid the state will be your own bank or credit card record.  Look at the filing instructions that print out with the pdf of your return and see how you arranged to pay your state tax due.

You must pay your state tax due using the state’s preferred method of receiving payment.  For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe

https://ttlc.intuit.com/questions/1901670-how-do-i-contact-my-state-department-of-revenue