Your state taxes owed were not paid twice. You can never pay taxes owed from a tax refund, federal or state. The only payment that can be paid from a federal tax refund is the TurboTax account fees and then only if you made that request before filing the federal tax return.
You will receive the full federal tax refund as reported on your federal tax return if you paid the TurboTax account fees by credit card.
A state tax due can only come out of your bank account or you have to mail in a separate check. At the end Turbo Tax just shows you the net combined amount of federal and state but that is only for your info.
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.