Nebraska

Mahoney is the best local to you for those ages, Indian Cave might be up your alley if you don't mind the drive. I'm not too familiar with out west but I'd be cautious doing that long of a drive with young kids on your first go around. Tips - bring lots of water, chose a more built up site for your first few times to guage your kids, get your park pass and reservation ahead of time, you can pack a lot on top of your roof though you will not pack most of what you pack after your first few times, bug spray is great but sunscreen is necessary, check for ticks when you are done, have some planned and unplanned activities for the kids, pace yourself.

If you're looking for something completely different but still in Nebraska, you can try Toadstool Park! https://www.fs.usda.gov/recarea/nebraska/recarea/?recid=10616

Although honestly, the Badlands are probably closer for you since you live in Omaha.

If you're into Bird and Sandhill Cranes, you can check out Rowe Sanctuary near Kearney, NE. Its not a park, but there are plenty of places to camp nearby.

I also really like the Wildcat Hills near Scottsbluff: https://outdoornebraska.gov/location/wildcat-hills/

If you're into Canoeing or Tubing, check out the Niobrara river. You can see the tallest waterfall in the state, Smith Falls! https://outdoornebraska.gov/location/smith-falls/

The state website for Game and Parks is actually really good, so you can check there: https://outdoornebraska.gov/

There's also a lot of places that are starting to pop up on Hipcamp that you can stay at. (Think AirBNB but for RVs and Camping) https://www.hipcamp.com/