Maestro,
Johnson City is a nice enough town and a college town (East Tennessee State U), but is not high on the list of tourist or traveler destinations, or so I think. That said, I have a longtime client who lives there, and about a year ago I met him for lunch at the Firehouse Barbeque Restaurant. It was good, even for an Eastern NC barbeque snob like me. Location is a converted fire station complete with a 1930s vintage fire engine inside. I'd readily go there again, and if I recall correctly, it's not far off of I-26, on the west side, near the ETSU campus.
Yes, I-26 follows the path of old US 23 for the most part, so if you're seeing I-26 changing over to US 23, down through Mars Hill and Weaverville, NC, that's the route of which I speak. I am 99% sure the last of the 2-lane stretches of US 23 is now 4-lane, but there could be a very few miles which aren't, and if that's the case, my recommendation does not change.
It's a nice drive to and off of the Blue Ridge enscarpment along I-40 east of Asheville to Morganton. My wife enjoyed downtown Black Mountain, NC when on a cycling weekend a year ago. Downtown Asheville is chockablock full of good eateries, brew pubs, and a contingent of local hippies, hipsters, and artists which would make San Francisco proud. Not exactly my personal cup of tea, but the people-watching is extraordinary!
I have never visited either Charleston, SC or Savannah, GA, so can not offer any suggestions.
Have fun!
Foy