Our favorite place is Pine Mtn RV Park in Pigeon Forge.
As many have indicated, it may be hard to find a spot this late in the game. The sites fill up quickly.
We have a site the weekend of Oct 20-22 at Douglas Dam Headwater campground (water/electric only) and we made ours in May. This is just outside of Pigeon Forge and right on the lake and is $25/night instead of $50 and 60 per night.
They do have 3 bath houses and the sites that are NOT on the water, but have a water view are nice. We usually stay in sites #14-20.
I just checked their website and they do have several sites still available for the first week and weekend of October. You will not be in the mountains, but you will be on the lake. They have usually dropped the level of the lake by October, but you still plenty of water to paddle around. We had both of our kayaks there last year and had a blast. The weather was perfect - warm during the day and cool enough in the evenings for a good fire.
Douglas Dam Headwater Campground