I think my first recommendation would be Tyler State Park.
Just a matter of local language, but anything "north" of Dallas we would call 'North Texas,' so in our speak all of NorthEast Texas would be east of Dallas.
There are several fine campgrounds on Lake 'o the Pines, not far from where I live; two near the dam are Brushy Creek and Buckskin. While Tyler State Park has full hook-ups the COE parks would have water, 50 am electric, and a dump station.
Daingerfield State Park would be a bit closer to the Arkansas border and is about 20 miles from our home. Since it is so close, I don't know about the hook-ups.
As for commercial parks there is Shady Pines near Texarkana. We bought our present trailer from Shady Pines RV and the park next door looks really good. Just depends on whether you prefer state parks (as we do) or the commercial variety.
There are also several COE parks on Wright Patman Lake near Texarkana, but I do not have any personal experience of staying at any of them.
Another state park is Bob Sandlin near Mt. Pleasant and Pittsburg.
All of these state parks I have mentioned are on lakes and in pine and hardwood forests. As you go nearer to Dallas you will lose the pine trees and most of the others.
Yes, it is hot and humid. We depend on air conditioners and we have two units in our fifth-wheel. Spring is the ideal time to visit East Texas and later in the Autumn.
I concur with the recommendation of Petit Jean in Arkansas. I would also mention Maumelle COE near Little Rock. There are many good campgrounds in the Hot Springs area.
Good luck and let me know if I can share information more specific to any of these campgrounds.
