Obviously, don't carry in schools (colleges included) or courthouses. Also, you will need a permit to OC or CC in Tennessee, whether on foot or in a car (doesn't matter which state; we recognize them all).
Now, regarding private property, Tennessee does give legal authority to businesses to prohibit carry on their property. The sign(s) that they post must be at every reasonable entrance and must meet one of two conditions: (1) it must contain specific wording from the statute (look up TCA 39-13-1359 if you want to read it), or (2) it must have a gun buster symbol (gun with a red circle-slash). If the business has one of these two types of signs, it carries the weight of law (meaning that, in theory, you can be arrested for carrying past the sign, regardless of whether the owner or person-in-charge of the business asks you to leave). If it has a sign that says weapons are prohibited but doesn't meet the above criteria, the business can still ask you to leave (and you could be the test case for what counts as legal signage).
Now, on to your main questions. I'm pretty certain Dollywood has a posted policy against firearms; but I'm not sure as to whether it meets the criteria I listed above for being legal signage. Also, I have seen few, if any, OC'ers from that area on this forum. However, if Pigeon Forge/Sevierville and Gatlinburg are anything like Knoxville or eastern Tennessee, I don't imagine you'll run into any serious issues.