I have read in trip journals that Grand Canyon NP does have HB sites, Zion does not, and am not sure about other NPs. Some pre-trip phone calls would clear that up.

Stealth camping should be no problem along the way, and I bet if you ask the police in most small towns along the way, they'll find a cheap or free camping option for you, often in the city park next to the pool. That's all part of the adventure.