I'd use UPS or FedX. If you mailed a pack it could take weeks to get there. A package that size and weight would go parcel post. Don't know where you'd ship it to though. Many motels will accept packages for guests. Maybe the Lodge at Amicalola?
Looks like you could take Amtrak from Auston to Atlanta or Gainesville. While this will take an extra day, you don't have to deal with airport BS and restrictions on what can fly.