Those areas have been my playground since the 70s when my father first took us up there in an old Franklin 11' TC. And now I'm getting my wife to fall in love with the area. :)
Wouldn't worry about late snow as it is rare and not a major issue. Worse case is you have to sit for a few hours until the sun comes out and melts it away. You do need to watch out for a few minor icy spots in the shade though. Not a problem, just be prepared. During the week the campgrounds will be almost empty and the weekends should be 1/2 full at most.
First, recommend you look at my photo album, then look at my trip reports most are listed in the TC Trip Report sticky. After that, ask more specific questions or my post will get too long. How long of a hike do you like? How secluded do you want to get? Any interest like gems, gold, grave markers, waterfalls, etc?
But some quick pointers -
Cataloochee for elk. If the campground is open, you must make reservations before you get there.
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Cades Cove for wildlife including bears, just be prepared for traffic. CG is open all year.
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Shining Rock Wilderness area parking lot for views. See my camper over there?
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Harmon Den Rd two miles north of I40 for convenience
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Most of the NF allow roadside camping as long as your are not impeding traffic and are 300ft from a paved road. Pisgah NF south of Asheville is the exception with camping at designated areas only.
When you get down this way, let me know and maybe I can meet up with you. I would love to pick your brain about places in your area and northward.
