Lots of time, Lots of Options
Welcome to the RTA Forum!
You could make the drive from NY to Dallas in as little as 3 days if you made it a speed run, so having 3 weeks will give you plenty of time to explore the East Coast.
It seems like you might be missing out on the mountains, so I might try to fit in some time at the Great Smokey Mountains or Driving the Blue Ridge Parkway. From there, you could cut back to the coast, and check out the beaches.
I also don't think you have to make Atlanta or the SC Beaches and either/or thing. With your timeframe, I think you could have time to do both.
Small Town Louisiana/Texas
May I suggest getting off I-10 when you are travelling from New Orleans to Dallas and take Highway 90 through Cajun country in South Louisiana. Then you can take highway 171 north from Lake Charles to Shreveport. And finally catch Highway 80 from Shreveport to Dallas. Those are all well traveled roads that have the added attractions of going through small town Louisiana/Texas, without ever being too far from a major interstate.
Laura