No you do not have to prebook campground sites unless you want to stay inside a major National Park. Since you have a generator you can stay at many National Forest (NF) sites and other no-utility sites. These are called boondocking sites by rvers.
Two weeks is not a lot of time so I suggest concentrating on a few parks rather than spending most of your time driving from one place to another. From LA I'd think about the North Rim of the Grand Canyon which has a lot of good NF sites right outside the park, followed by Bryce Canyon which has good commercial campgrounds right outside the gate with a shutter services, and Zion NP where you can camp outside the park. Spend 3 or 4 days at each. With drive time that will use up your time and allow you lots of time to explore these interesting parks.
In my experience once you are away from major metro areas you will find that that the campground crime rate drops to near zero. In 30 years I have never experienced any sort of criminal activity while we were camping. I think you'll find most people here have had the same experience. If you stop at Walmarts or other overnight parking locations then you need to be a little more careful. If the store has armed guards then probably it's a good idea to move one. I've only seen this a couple times, once in Homestead FL.
I've never done this but others have frequently mentioned it. You may want to check into flying into Las Vegas, which is supposed to be very cheap. And renting your RV there. That would add 3 or 4 days to your actual vacation time and allow you to add another park. Maybe Arches
The best thing for you to be doing right now is this: get a good road atlas, study the maps of this area and look at the area you want to go to. There are a number of smaller parks you can visit on the way, Coral Pink Sand Dunes SP is one example. Visit the national park website and look at the parks. You really are thinking of doing a subsection of the Grand Loop which is:
South Rim Grand Canyon
North Rim GC
Zion
Bryce
Arches
Capital Reef
I suggested the North Rim since it is closer to the rest of the loop and is less crowded, but lots prefer the mob scene at the South Rim.
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