Unfortunately, the option for giving a date for the payment to be made only appears when you're in the filing process. Once you file your returns it cannot be retrieved and, as you noticed, it isn't listed anywhere in your return record copies.
All you can do now is try to remember what date you gave and check the account you setup to make the payment from after that date (note that some banks/credit cards may take 3-5 days for it to post).
BTW, TurboTax does not pay the state taxes. The payment account information is passed to your state's tax office and they submit for the payment.
