Jon and I use this app to help find most of our campsites when we are in the U.S. It lists most of the places to camp on public lands. Some are really out of the way places to camp and others may be at a city park in town. This includes dispersed camping, municipal sites, Corp of Engineers dam areas, military bases, and state & city campgrounds.
The app downloads the basic information on your phone so even if you don’t have cell coverage, the basic information is still accessible.
Some of the campsites are free, especially the ones that are dispersed camping in BLM land. Even the ones that do charge are usually less expensive than commercial campgrounds especially if you have a Federal Access pass. Some places may also require paying for an entrance or day use fee in addition to the campsite. That is especially true in some state parks including ones in Texas and Colorado.
Please note that some of the campsites require a high clearance or even 4 wheel drive vehicle to reach them, especially those in BLM or Forest Service land.
As usual, I wouldn’t recommend that you base all of your plans only on this reference since district policies and prices do change on occasion. You may also want to bring small bills since you often are unable to pay with a credit card or get change.