TurboTax did not pay my state directly, so I went ahead and paid the state. Just saw that TurboTax NOW paid also. How do I get my money back for duplicate payments?

TurboTax does not make any sort of payments for you.   TT only passes the payment information on to the IRS or to the state and the IRS or state withdraws the payment.   TT has no control over when that will happen, and in fact, TT never even knows if the payment was made.    Contact your state to see how to get the duplicate payment back.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/state-taxes/contact-state-department-revenue/L9qVToi02_US_en_US?uid=m6e06um0