There is not much in the way of camping near Raleigh. We did discover Jordan lake
has some nice state park CG that have some waterfront sites and beaches on the lake to swim in. Be aware that state parks in NC close and lock their gates from dusk til dawn. You must be back in the park by dusk or you will not be able to get to your rig without a really long walk in the dark. Crosswords and poplar point are the better ones. They have ranger programs for the kids too.
I have been wanting to try tanglewood park
in clemmons nc, not far from Winston Salem. Another great area is Boone NC whee they have a railroad you can take the kids on.
When you make your way to Asheville check out Hooker Falls. You can hike to the falls and go swimming there. We swam below the falls but there were people jumping off the falls. The water was very cold but it was fun.
A good beach CG is Camp Hatteras in rodanthe.
For places to eat... kinda out of your way but Dickie Dos BBQ in Haw River is phenomenal. If you stay at Jordan lake, you will be near cary and apex. When we lived in Cary, we loved Ginza (Japanese steakhouse). Danny's BBQ in Cary is also good, although Dickie dos is far better and Hillsborough bbq is somewhere between those two.