These are, for the most part, routes that are familiar to me. I will give you one possibility that fits into the states you have mentioned.
Of course, I-20 to Shreveport, but you could take the South loop around to I-49. One good possibility for overnight would be Poche's Market and RV Park at Breaux Bridge, LA. Great Cajun food at their restaurant.
Continue from there on I-10. Several possibilities for overnight:
Buccaneer State Park in Mississippi, Gulf State Park in Alabama, or best of all, Fort Pickens Campground National Seashore near Pensacola, FL.
In Georgia there is Blyth Island Regional Park near Brunswick, GA. From there you have Savannah, Charleston, and Myrtle Beach. Make your own choice.
Then on to Garner, NC. We have stayed at all the parks mentioned and many more along this same route. All great choices.
EDIT to add: I live 20 miles north of Longview. Will be glad to answer any specific questions about this route or my suggestions. Please use a PM (Private Message) if I can be of more help.