We never had issues with stopping at a state park of national forest just coming off the road. But we call during the day to find out the late arrival procedure. I will normally take a list of such parks so I know where they are. I usually check to see if there are a lot of sites available before I leave.
I have stayed at a walmart once and rest areas a few times. Worst place to try to sleep. Bright lights and much noise.
The other thing we have done is stoped at free campgrounds. But they fill up quickly by about 9:30
https://freecampsites.net/