If you are 62 or older be sure to get a "Senior Pass" from the US Forest Service or Parks Dept. $10 for a lifetime card that gives you 50% off at five federal agencies -- National Parks, National Forests, etc.. Get info at their website: America the Beautiful - National Parks and Federal Recreational Lands Pass
With the card you can get half price camping at Bureau of Land Management and Forest Service campgrounds -- $12 to $15 a night is a steal. I have a book that lists all the BLM, Forest Service, Fish & Wildlife, etc. campgrounds in the country. You should be able to find one at your local bookstore. If you can plan your trip around these campgrounds you can save a ton of money.
I have never had trouble getting spots except an occasional Friday or Saturday night. Weekends are busy but Sunday night thru Thursday nights you can usually find a spot.
Oregon has great state parks. They raised their rates this year to (avg) $30 per night. (little less without sewer hookup -- little more with) You can download a state parks brochure here:
Have fun. We made a similar trip in reverse a few years ago -- Portland, Oregon thru Utah, down to Carlsbad, over to Dallas and Houston, up to Memphis, then Chicago and Indianapolis and finally home. You'll love it.