Have not stayed at Natural Falls. What is your route?
If US-59, I can recommend Honey Creek SP in Grove, as well as Cedar Oaks RV Resort (our usual place on Grand Lake). Grove has several other commercial RV parks.
Also on Grand Lake but a bit off the route for US-59 are Bernice SP on OK-85A (west of the lake), and Twin Bridges east of 59 on US-60. Both are somewhat remote from local attractions or amenities, compared to Grove. I've stayed at Twin Bridges a couple of times, and visited Bernice SP several times taking a "shortcut" into Grove (coming from the northwest). Campsites at Bernice are on the lake, while Twin Bridges has a small "fisherman's camp" on the lakefront with the rest of the campgrounds up on the bluff.
If you use OK-10 to go north from I-40 there is a really nice SP at Greenleaf Lake (between Gore and Braggs), a nice Corp Park at Strayhorn Landing on Tenkiller Ferry Lake (a bit east of 10 on 10A), Sequoyah SP (to the east of Wagoner on OK-51), then ultimately you connect with US-59 going into Grove.
Basically, the whole area between US-59 and US-69 is filled with reservoirs on the Arkansas River system, each with several Corps parks and in many cases state parks. This thins out as you get into Kansas, but there are still places to stay. Along US-169 there is the Walter Johnson (municipal) Park north side of Coffeyville or the Santa Fe municipal park at the south edge of Chanute. The first is about $10 a night, the second was free for your first night, last time I checked.
The other big section of camping in Oklahoma will be where the Arkansas River swings west and north out of Tulsa, going toward Wichita. Most of these are west of US-75, east of I-35, on Keystone Lake, Skiatook Lake, and Kaw Lake. Headed toward Omaha, however, I would consider these areas to be well out of my way.
I've not checked out what might be along US-75 in Kansas, as my usual route north is 169. My experience is that most counties in Kansas have a small fishing lake, at which you might park an RV, but will have no designated campsites or utilities.
Along US-75 in Oklahoma there are state parks at Okmulgee Lake and Dripping Springs Lake, a RV park in Bartlesville (full of snowbirds just now), and a Corps park on Copan Lake (which has a small non-reservable year-round RV section at Washington Cove.
There is also a question of "when" because on some of the possible routes there are a number of nice Corps parks, but those are mostly closed to camping until the volunteer hosts start showing up in March. At the state parks in winter you might find campgrounds open but water may or may not be turned on. The commercial parks in Grove stay open through the winter.
If you access Oklahoma state park information on travelok.com you will see a green button "Book RV Campsites." That will usually lead you to believe there are no campsites available. What it really means is that there are no campsites to be reserved, because in most OK SP campgrounds the RV sites are all "first come, first served."
