Does Turbo Tax give you the option to pay taxes owed to a state out of your federal refund when you are completing filing? How do I find a record of this happening?

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.

No, TurboTax does not provide that option.  You have to pay state taxes with a check or credit card.

No...Never.

You cannot choose to pay your state taxes by auto-directing some of the Fed refund.

You must either get your Fed refund yourself, and pay the state yourself separately...or just pay the state as soon as you can ignoring what the Feds "Might" send you.

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There was a page that showed you the "net" of what the final effect of refund vs. payment was...but that was an Info-only screen to perhaps make you feel better.  (Yeah, so many people misinterpret that page, that TTX needs to get rid of it)

i.e.  the state will never wait for the Feds to take action and pay them.  Sometimes the Feds will decide you are not actually due a refund....so the state won't sit around wondering when they'll get paid.  Thus, the auto-direct of some refund to pay the other is NOT allowed .  Neither Feds not States want to get into all the eventual finger pointing.