Nutmegger

Pulaski

That'd be a long haul, even starting in Atlanta, but from Texas you could follow the Natchez Trace parkway from Natchez, MS to Nashville, then cut over to Bryson City and follow the Blue Ridge Parkway to Rockfish Gap in Virginia. Those two routes would total over 1,100 miles, which IMO is plenty for a week, especially as you have to get to Natchez, and if you then also drove the return leg you'd presumably be looking at 2,500 miles, which I am not sure would make for a relaxing 7 days. ... In a rented car you could head north from Rockfish Gap and fly home from DC.

My thought was the Pacific Coast Highway.

Hotscot

Cliff on one side.
Death drop to the sea on the other.
Can't look cause you're navigating.
Nose to tail because gawkers.

Most boring drive in America.

395 where it's at.

lizzyq

Watch out for a '97 LR Discovery and a middle-aged jeep driven by our son and his classmate from NMT, the disco may be short of an attached exhaust

Rete

Natchez, MS is lovely in May. Enjoyed our weekend visit there last spring. You can cut across from LA via bridge over the MS River. As said the drive up to TN isn't far and I can do it within a few hours from Hattiesburg, MS. How long a bike ride would it be to the SC coast? Charleston is someplace on my Bucket List as is some of the towns in TN that I have childhood friends living in for decades now, i.e. Gatlinburg, Fort Knox.

Pulaski

Charleston is definitely worth a visit, and a rare example of an American city that is truly historic, but the SC isn't much - it is mostly marshy swampland with inlets, so the "coast road" is miles from the coast, flat as a pancake, and lined with pine trees.