The ACA maps indicate which towns have known camping areas. Some may be a city (paid) campground while others may just be a spot in the local park with no services.
I personally always ask for permission so I am not ask to move in the middle of the night (happens occasionally if you do not ask permission). In order, I ask the 1) fire department, 2) library, 3) police, then 4) city office. I have found the fire department tend to be the most helpful and occasionally you can get a free shower from them if the city park does not have a shower or a city pool (which has at least a cold shower). If this does not work out, I ask at a local church but in small towns, the pastor may not be in the church during non-Sundays. Wherever you stay, always leave the site better than you got it so cyclists do not get a bad reputation so then others are not allowed to camp. Also, if possible, buy something at the town's cafe, convenience store, etc. if the camp area was free.
Since the TransAm is so old, I would think most permitted places are indicated on the ACA maps but I could be mistaken about this. If not, they should be!
Don't forget about WarmShowers. If you do not know about this group, it is a reciprocal hosting group (like Couch Surfing) but ONLY for touring cyclists. If you sign up, you are supposed to at least consider if you can host if someone contacts you. Unfortunately, a LOT of people sign up to receive but then blow it off once their tour is done. Be sure to read the reviews of the host and be sure to give appropriate honest feedback if you use it. And be sure to return the favor and host others passing through your town when you get back.
Tailwinds, John