Your second plan is much more do-able. You will be able to guess my gender when I suggest that you build in a little time in some of those private campgrounds to do laundry. Yep, one of life's little necessities that takes anywhere from 3-5 hours out of a camping trip. (It can be done while you working on dinner, or if you want to sleep in shifts. :) )
Remember to MAKE ADVANCE RESERVATIONS in state parks and national parks particularly. People do show up & find campsites, but if you bank on that, you may find yourself being turned away in Yosemite, Yellowstone, etc.
BTW, I know you're aiming for least expensive, but the CG you chose for the Badlands is basically someone's backyard - or so it seemed to us when we drove by. I'd recommend you look instead into staying in the national park CG (they have recently installed electric hookups, and you WILL need air conditioning there) or at the KOA, several miles south, which has shade trees as well as hookups. The Badlands can be "done" by driving the loop, which starts and ends at I-90, and unless you want to go hiking - remember, in summer, this area can be upwards of 100 degrees - you may be content with just driving through. Just a thought. That way you could add an extra night up in the Black Hills (more pleasant) or somewhere along the way across SD. There may be some Corps of Engineers CGs along the Missouri River, mid-state in SD, which could be somewhat inexpensive.
Walmarts can be somewhat few & far between in some of the western states. Not sure how many there are between Yellowstone & the Black Hills (Mt. Rushmore) - Cody has one and I'd guess Buffalo, Wyoming might have one; perhaps also Gillette (WY) or Spearfish (SD). That would be about it. The direct routes between Yellowstone & the Black Hills cross the Bighorn Mts - slow going, and make sure to watch engine temps on the way up and brake temps on the way down (experience talking)!