Tuckaleechee Caverns is a really cool place to visit, been there several times, Cades Cove is beautiful too, especially in the fall!Not only do they not prohibit carry, they openly welcome concealed carry. I asked them that question a couple of months ago.
Also, keep in mind that No Carry signs themselves do mean something legally in TN, unlike GA.
If you are in that cheesy tourist trap town, the aquarium is a must see. So are the Tuckaleechee Caverns, about 30 minutes away. That was $18 or $19 very well-spent, especially if your tour guide is the guy that has worked there for 18 years, is extremely knowledgeable about the cave, and is a very enthusiastic speaker.