I just found it! Route 119
So it's not US-52 that we would take into Charleston, it's US-119. According to Rand-McNally, which is where I am using their roadtrip planner, the drive from Corbin, KY to Williamson, WV is 159 miles at 3 hours 35 minutes and the remainder of the drive to Charleston, WV from Williamson is 88 miles at 1 hour 55 minutes.
Have any of you driven US-119 from Kentucky into Charleston? It looks like a fairly windy road, but could potentially be a gold mine for scenery and cultural experiences. From the limited amount of research I have done, this area seems to be rich in coal history. Old mining towns would be so fun to see.
Please let me know anything about this route! Thanks!