Rvparkreviews.com is helpful too - but of course you need to know your route(s) first, so you know where to look.
If you're leaving that early in the season, you might want to take a more southern route out and northern one back. However, I-70 goes right through the Colorado mountains, so it won't have any advantages of "earlier spring/summer" than being farther north.
At that time of year it will be chilly at night in the areas you mention, and likely still below freezing in Yellowstone (overnight), so be sure to be prepared for that. Given your schedule, I'd definitely recommend going to the Grand Canyon first, then head north to Yellowstone and back through the Black Hills (Mt Rushmore, etc.)
If you're not traveling on a weekend, you might consider state parks, or if you're on I-80, some of the "waysides" in Nebraska for overnight stops. More nature than the private CGs and some have electric hookups. We camped in Kearney State Park with a 34' TT and had a site with enough room for us. Weekends, the public CGs tend to fill up, especially if they're near a city.
2020 Toyota Tundra CrewMax 5.7L V8 w/ tow pkg, Equal-i-zer
2020 Lance 2375