Refunds on The card was immediately charged when it was supposed to be debited at check-in and now it says I never paid.

I made a hotel reservation without charge for cancellation totally refundable and I canceled the reservation 33 days before the deadline. They say I never paid and I will not refund money. It was clearly loaded on my card. I wrote to and they said they would have to open an investigation that will take 2 billing cycles. Has anyone experienced something like this?