Most National Parks do not have shower facilities. Check in detail for each park, though. And most of them have limited spots with full hookup.
State parks are highly variable---no way to know except looking at the park website.
Almost all privately owned RV parks will have electric and water, some have limited sewer. The quality of the facilities is EXTREMELY variable. Look on RVParkReviews.com for info on any privately owned (and some public) facilities you are interested in.
There are a couple of categories you didn't mention:
US Corps of Engineer Campgrounds---many of these are very nice, usually have limited amenities.
National Forest Service campgrounds--often in very very nice locations, and usually pretty inexpensive. Amenities range from full hookups and big shower facilities, to zero.
Many areas of the country have abundant city and county camping facilities as well. They are usually inexpensive, but again amenities are highly variable. Most of those seem to be in the Midwest & plains states.
You can find all kinds of info about all publicly owned campgrounds on a site called uscampgrounds.info. It's a great site. Very useful for trip planning.
I find that having internet access while you travel is CRITICAL to finding a place to stay. The best way to get that reliably is with a 4G wireless hot spot. Check out Verizon or ATT for the best coverage. Pick a device and then go to mobilekarma.com for a good deal on a used one. You can prepay it and keep it working for just the months you are travelling if you don't need it otherwise.
