When I am travelling across the country like the OP proposes, I use Google maps and look for little green spots along the route. Might be a state park, county or muni park, COE, NF, whatever. Do a little research on the spots that look interesting and stay at places like that. Lot's of great places to stop I have found with that method, some hidden gems. I do tend to avoid private rv parks, so YMMV. Weekdays, most of those parks have plenty of space to just show up.
Everyone's idea of leisurely travel may be different, but I like to start after breakfast, maybe 10-11 and be done by 4 or 5. 250-300 miles. Avoid interstates and see something along the way. Once you get much west of the Mississippi, the 2 lanes are pretty easy driving.
I totally understand those on a tight schedule. Been there and done plenty of 800 mile days. Given a choice the slow ride is a lot more enjoyable.