How can I resolve this? I received a call from the IRS saying I still owe $68 to the state but it should have come out of my refund. I paid the $40 fee. Help!

No. State tax payments Never come out of your IRS refund. They are your responsibility to pay when you file.

I don’t think you understand. There is an option to pay the state tax with your refund for $39.99 on TurboTax when you file.

Nope; Option applies only to paying TurboTax fees out of Federal refund.

The IRS does not call you.. And the IRS does not care about your state tax.  The IRS is only concerned with your federal tax.