We are a little over 2 hours from both Asheville and Pigeon Forge/Gatlinburg areas. We enjoy them both.
Asheville should be good for 2-3 days. Biltmore is there and that can take up a full day. It has a large art and crafts population and you should be able to take in a few shops that offer handicrafts. We have never camped in Asheville, but friends of ours like Mama Gertie's.
I would NOT pull my TT across the Blueridge Parkway. Park at a campground and use your truck to see the sites. If you aren't used to pulling on small two-lane, curving roads, you should definitely NOT use this time to try it.
Pigeon Forge/Gatlinburg can be a great place or can be living H#*L! There are lots of shopping in the PF area. Gatlinburg has lots of really nice small shops in town along with OberGatlinburg (ski area). There are also some not so nice shops that sell, well,junk! Really good restaurants abound in both towns.
We really like Pine Mountain RV Park in Pigeon Forge. It is a "resort" and sites are smaller, but so are most of them in the area. It has a couple of pools, indoor and outdoor, so you can use them in May. It is a couple of blocks off the parkway, so you don't get the noise that you do in many of the ones ON the parkway. It is in main area, so you don't have to spend lots of time driving in the PF area. Sometimes (weekends and holidays, especially), traffic can be horrendous and it will take you hours to be a few miles. During the week, it's not usually as bad. On the Pine Mtn end of PF, there is not as much traffic. It is closer to Gatlinburg. It, along with the rest of the campgrounds in Pigeon Forge, has a trolley stop so you don't have to drive. Dollywood is good if you have kids and May is usually a great time to go. It is kinda $$$, but it's an amusement park.
Elkmont Campground, in the GSMNP, is a really good campground, but it is dry camping. The fireflies are there the 2nd week of June (give or take) and they are magnificent. Google them.
If you are looking for another place to check out, try Natural Tunnel State Park in Virginia. It's about 3 hours from the PF/Gatlinburg area, heading back north. We will be camphosting there in May and would love to meet you and show you the park. Views and the "natural" tunnel are magnificent and it's a great state park and campground.
If you have any specific questions, PM me.
2017 Ford F350 Crew Cab 6.7L 4x4 DRW