Should I still be getting a tax bill when I paid what I owed through TurboTax? Did they ever pay it for me? I opted for it to be taken out of my Federal refund.

Sorry you misunderstood.   You can not have a state tax due deducted from your federal refund.  NEVER.  You can only have the Turbo Tax fees deducted from your refund.  See,  Can I pay my state taxes from my federal refund?

Your only proof of payment is your bank statement or credit card showing the payment coming out.  Turbo Tax only passes your bank account number to the IRS or state for them to take it out of your account.  (Most states do not let you pay by Direct Debit so you have to mail in a check or go directly to the state's website to pay).  And some banks use different routing numbers for deposits and payments.  Double check the number you entered with your bank.

Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.