On the Trans America you will find that especially in the middle of the country the trail has been blazed a bit for you. Town parks, picnic areas, church yards, and firehouse lots have been scoped out in many cases and are listed on the maps.
After using the maps a while you get a feel for what will and what won't fly. I found that I could find places to camp in town and not be hassled on my own pretty well in other towns not on Adventure Cycling routes especially in the middle of the country. I found that the people were open, kind, and generous if I was the same. This despite the fact that they usually didn't approve of my politics or my spandex neither of which was usually mentioned.
I personally most often camped in town in the great plains in tiny towns. I'd generally buy some of the stuff I needed any way at the general store and not actually as permission, but say something like "I am biking coast to coast, do you think anyone would bother me if I slept in the picnic area over there tonight?" I find that leading with the biking coast to coast thing helps a lot. Being open, friendly, and making eye contact (always take off sunglasses) is important.
Sometimes I do ask at the general store if there is a place I can camp for free. I might ask other folks that I meet the wait staff at a diner, whoever might have seen others camp or know of a place. The local police station is another, but I don't to often stay in towns big enough to have one, libraries are another place I have asked if they knew of anywhere. Librarians can be a font of info and will often call around all over the place looking for someone who has a spot for you to pitch a tent.
I will just add the I was really impressed at the kindness and generosity of the folks I met in the great plains, eastern Colorado, Kansas, and western Missouri. They were amazing. My first coast to coast trip kind of restored my faith in humanity.