TurboTax never pays your tax due. Only you can pay your tax due, either by sending a check or money order to the state, by paying the state on the state's tax web site yourself, or for some states, by entering banking information to allow the state to take out a direct debit from your account. Do you see those payments being taken out of your own bank or credit card account more than once?
TT leads prompts one to send the appropriate info, in my case the bank numbers for the transfer. This happened two times, and the state debited my account twice. I can see it in my bank statement.
For Arizona, if I remember correctly, besides the actual tax return, there has to be an additional communication to the state giving the link between the return and the account info - something like that. I'm thinking that I need that information, plus a lead into the Arizona Dept of Revenue to be able to initiate a refund.
I haven't been able to find either of those pieces of information.