Almost the same thing happened to me last Summer with Budget. Had booked a non-prepaid four-day rental at RDU, found a cheaper rate with Hertz and went with them. I neglected to cancel the Budget reservation. My bad as that likely would've avoided the whole situation. Not only was there a charge for the entire duration, but the pick-up and drop-off times were times I wasn't even in North Carolina.

I contacted Budget/Avis corporate and they put me in touch with their Security/Fraud department and they are the ones that handle these situations. The agent was very helpful and handled mine quickly. They discovered it was simply a clerical error as they applied my no-show reservation to another renter. Got refunded in full within a few days.