I feel like I am highjacking the thread as I have no advice on where to camp in the areas mentioned in the original post. I see lots of folks telling of their Route 66 adventures, though, so here goes.
We tackled about 250 miles of Route 66 from Illinois to Indiana in 2010. This was in my
1962 Corvair Greenbrier camper van "on the way" to the Corvair national convention in Cedar Rapids.
We enjoyed lots of awesome sights and experiences and I was thrilled to drive those miles on the Mother Road in a true vintage vehicle. We did camp one night in Illinois and enjoyed the monotone voice of the weather radio all evening as there were tornadoes touching down in every county around us. Good times!