I spent two spring & summers in those states without reservations, and the only problem I had was getting into most of the Oregon coast state parks (it did take a couple of tries to get a space at Dead Horse Point State Park in UT). Other than that, I usually found a place even though Reserve America often showed the parks full. They don't list the first come non reservable sites. If I wanted to get into a popular campground such as the south campground at Zion, I found you need to arrive early, usually before 10:00AM.
Some of the National Parks post fill times, both current & previous years that can be helpful for planning. For example, here are the fill times
for Apgar at Glacier National Park.