One of our favorites is Twin Bridges State Park near Miami. From I-40, take I-49 north to Springdale, US 412 west to Kansas (that's the TOWN in Oklahoma), US 59 north to US 60, east to Twin Bridges. It's first-come, first serve, rarely completely filled.
Another is Keystone State Park west of Tulsa. A third is Sequoyah Bay State Park near Muskogee. I like to visit the submarine (!) at Muskogee. (That's right, a WWII submarine, the Batfish, in Oklahoma. I've visited four submarines and the Batfish tour was the best of the four.)
All of those would be possible, being north of I-40 and east of I-35.
