Based on what you have said, this seems like an obvious mistake by the Georgia DOR. It looks like you have to write back to them, telling them how and when you made the payment, and attach a copy of the confirmation that you received. I would also include a copy of the transaction in your bank account records. Follow the instructions in the letter for how to reply.
You probably don't need any help to reply to the letter. But if you want some help, and you purchased Audit Defense, call TaxAudit at 877-829-9695, or report the letter on the TaxAudit.com web site. They will help you with the letter, even though it's not actually an audit. If you did not purchase Audit Defense, you can get help with the letter from TurboTax at the following link. The screens talk about an IRS letter, but they will help you with a state letter also.
We can help you with your IRS letter
