You misunderstood. You cannot pay taxes owed using a tax refund, federal or state. You can pay the TurboTax account fees with your federal tax refund, nothing else.
Your only proof of payment will be a cancelled check, bank or credit card account showing the payment, depending on how you selected to pay the taxes owed in the File section of the TurboTax program on Step 2.
You need to pay the state taxes owed as indicated on the letter you received from the state.
