I received a refund from Federal but ended up owing state. From my understanding the amount owed to state was taken from my federal return. I received a letter from state today stating the amount owed was never received.
Your understanding was incorrect. Federal and State results are Never combined for action, only as a matter of Information. If you owe State (or IRS), they are Not mixed for you. You must pay what you owe, and accept any refunds, as independent actions.
Your state tax due cannot be paid from your federal refund.
It is not possible to pay your state tax due from your federal refund, nor is it possible to pay a federal tax due from a state refund. The federal and state treasuries are completely separate from each other. You may have seen a TurboTax summary of your federal and state amounts that confused you into thinking one was “paying” the other. It is not. If you have federal tax due you can pay by mailing your payment with the 1040V voucher, (which has the address printed on it, having the payment taken out of a designated bank account, or you can pay directly on the IRS website.
You must pay your state tax due using the state’s preferred method of receiving payment. For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.