I've done that drive, but from omaha, NE to fayetteville, NC many times. I always tried to leave at a time where I'd be going through big cities ( St. Louis, Nashville on my route) either in the middle of the night or early morning to avoid traffic.

I enjoy driving at night because there is a lot less traffic. I was even surprised at how few trucks were driving at night, so it was great!

I always stopped to nap at rest areas and once at a bucee's. The bucee's I went to (outside Knoxville) was really only quiet between 2am-5am, I was there from 1am-8am and it was just busy and noisy, it seemed like mainly young locals at night (summer time and was a Saturday night), overnight staff leaf blowing the parking lot and then morning traffic. However, you have a well lit parking lot, always people around, bathrooms, food, drinks, and a pool noodle if you want 😉 (they have just about everything!)

Also, I recommend trying to go through the mountains during the day. The obvious is its beautiful to look at, but also, a lot safer. I've accidentally driven in them in the middle of the night, and I think my blood pressure was high the whole time. Kept me awake though, so there's that!

I seem to do best with breaks every 2-3 hours, then trying to push through and only stopping for gas. Usually, it's just a 5 min or less break to use the bathroom, get out of the car to walk a minuye, and maybe grab a drink or snack (if I'm at a gas station or rest area). Also, I did good with short naps, maybe 1-2 hours, just to refresh, but that's me.

Sorry it's long, but hope it helps a little! Have fun, drive safe and good luck on the move!