If you like traveling slowly and enjoying the scenery you might try the Blue Ridge Parkway. There are many options of things to do along the way, from winery or brewery tours, golf, historic sites, wildlife viewing, hiking, canoeing, music, crafts, shopping, etc. I like them, but the NPS campgrounds do not have full hookups, though there is always another option nearby. This time of year, in the NC portion of the parkway, it will usually be in the 70's during the day and 50's at night, though it has been cooler than normal for the last few weeks. I hear Virginia Highland Haven is a good place to stay, and it's an easy one day drive from you. Three of my favorite locations to visit along the BRP are the areas of Blowing Rock, Asheville, and Brevard.
2014 Silverado 1500 5.3L maximum trailering package (yes, I'm towing the 34')