I rode the TransAm from its start in VA to about 100 miles into Colorado. (I had to end it for personal reasons, which really sucked.) I hope to do the entire thing some day.
Campsites range around $12 to $20. Past VA, you'll find some city parks in Missouri, Kansas, and KY, and in some other states. You can stay there for free, and these places often have a swimming pool and showers and picnicing gazebos. The managers see touring cyclists often, so you won't be a surprise, and they're generally welcoming. Just be careful where you pitch your tents. These parks can get very active and noisy in the summer: baseball games, tennis courts lit up bright until midnight, etc. If you're a light sleeper like me, you'll want to camp as far away from that stuff as possible.
As you apparently know, the TransAm Maps from Adventure Cycling will help you find campsites, motels, etc. Some hostels and bunkhouses along the route are great and cheap. In Damascus, VA, for example, the Mt. Rodgers Outfitters bunkhouse is only $10. And it's a cool little town.
The motels in VA can be expensive ($50 or so) for a cheap but clean room. Things get cheaper in Kansas and MO.
Good luck on your trip! I'm dying to go again, maybe in 2008 if I get the balls AGAIN to quit my job in order to do the trip.
David in Florida