I rode the Tennessee section of the trace in 2009 as part of a transcontinental route. Stealth camping on the trace is illegal but definitely possible. The old trace, which is basically doubletrack, intersects the new road at regular intervals. I woud recommend, going down the old trace until you are mostly out of sight of the road, and set up a tent there. As always, wait til dusk to make camp.
I have had some trouble with critters (skunks and raccoons) getting into food bags before, so hanging your bags is probably a good idea. Neither of those animals was able to rip open my panniers, but a skunk was smart enough to open a poorly-closed ortlieb roll-top.
I didn't see any other tourists, but I ran into an awful lot of roadies when I got close to Nashville.