We're with you in not liking reservations. If you're doing quick overnights along the highways, you'll get a site.
For longer term such as in national parks, look at the official national park websites for their campground information. You'll find that some campgrounds within the parks don't even accept reservations. Those are the ones we aim for and have succeeded in getting a site even with our 40' motorhome. For instance, Apgar in Glacier, Mammoth in Yellowstone, Colter Bay and Gros Ventre campgrounds in Grand Teton Nat'l Park are ones we've stayed. You do have to stay nearby the previous night and move into those campgrounds early morning.
Other times we've gotten cancellation sites in the major parks...again by being there early morning.
Also, don't try for the center of the park. Lewis Lake in Yellowstone usually has openings later in the day
There are also plenty of national forest campground which are beautiful. Baker's Hole a mile north of West Yellowstone is one we've stayed multiple times. No reservations accepted so get there in the morning.
If you're not fussy as to where you stay your plan is doable. Have a great trip.