http://gastateparks.org/ Look at Amicalola Falls or Cloudland Canyon...both are beautiful
http://www.recreation.gov/ This takes you to Corp of Engineer Parks...some have cabins.
http://www.tennessee.gov/environment/parks/findapark/cabins.shtml And of course Tennessee has some of the nicest parks around. I like Standing Stone - not crowded, great cabins, good hiking trails.
www.vrbo.com for private homes, condos, cabins all over the world.
Fall Creek Falls is nice. They have several campgrounds. I have camped there with dogs (in an RV) and had no problems. They have several hiking trails with varying lengths. Check out the TN state park website for more details.
Thanks everyone for all the information. I have checked Fall Creek Falls and they are booked for the next 3 months. I am going to check into Pickett State Park and see what they have available.