We've full-timed for 16+ years and don't make reservations unless we really need to be somewhere on a specific day; i.e. visiting family.
We travel constantly and public parks are always our first choice. I can't recall ever being turned away.
As full-timers your have the opportunity of getting settled mid-week for the upcoming weekend. Parks don't have reservations on every single site. They always keep some open for those dropping in.
Even for the big 3-day holidays we get a site without reservations. However, we then make it a point not to stay in the holiday 'preferred' areas - no lakes, rivers, theme parks,etc. We find a spot off the main drag and enjoy the quiet.
We even go to the major national parks - Yellowstone, etc. - and get a site in the non-reservation campgrounds. You just need to get there early morning when folks are leaving. It's really never been a problem for us.
It's a different story if you're still working and have limited vacation time.