My travel speed is also about 62 MPH. I drive a 38' Class A diesel. I find I do have to stop every 2.5 to 3 hours to stretch and walk the dog. Weather can be severe in the gulf coast and may slow or stop your travel.
There is a KOA right off I-10 at the Milton FL exit which is just east of Pensacola. They have a pool so kids can go jump in and burn some energy.
The campgrounds do seem to get a bit harder to find in the AL, MS and LA states. They are there but typically further from interstate.
Houston traffiic will definitly slow your average speed down.
The trip can be done as you planned - but it will make for some long days. When I drive my car (not the RV) I make Tampa to CA in 2.5 days but that is 14 hour days behind the wheel.