If you want a great place only about 90 minutes from Kansas City, I would recommend The Outlet at Melvern Lake in Melvern KS. Just a couple miles off I-35. It is corps of engineers park, very open and relaxing.
From there you are only about a 45 minute drive to the National Tallgrass Prairie /Wildlife reserve near Strong City KS. There you can hike in the prairie and (if you are lucky) see one of the few places in the lower 48 where there is a heard of free roaming buffalo. (no pets allowed in the hiking area).
https://www.google.com/url?sa=t&rct=...RgShpMT_v7NFYQ
Also note that it will be humid and warm enough that you will want to be able to run your AC that time of year, so you will want electric available.
Another nice place to camp nearby is Canning Creek in Council Grove. Also corps of engineers. On a lake so you get the water.
In eastern part of Nebraska there is not a lot that I can recommend from experience. I hear the state park by Nebraska City is nice. And there is a waterfowl area near Mound City MO that is nice if you like that type of thing.
In Missouri in the KC area, the state park at Weston is really great. Small so you would want reservations. On the bluff of the river, so a nice hike. Fenced in dog park. Easy to drive into Weston (small town) or 15 minutes to KC for dinning.