When you get a rough route planned out post it and ask for specific suggestions on where to stay.
Based on what you said about not wanting to be in a trailer park I would caution against the KOAs. To me they don't quite have the camping feel as they are too crowded. However, kids being kids, they are great if you want to give them a pool experience.
The national parks do tend to fill up fast, particularly for the larger sites. Try to get reservations at your most desired parks and then plan the rest of your trip around that. But if you can't get them don't despair. When we stayed at Trailer Village in the Grand Canyon (for example) the entire stay you could drive up and get a spot. However they never took the park full sign down. You just needed to know to ask.
In the west there are national forest campgrounds that don't take reservations. If you know where to look or just follow the signs you show up, fill out your envelop and get a sight. Need to keep small bills on had to pay cash. Here is a place we stayed at when visiting Yellowstone with no reservations.
1994 Brave 29RQ RV: National Forest camp ground just N. of Yellowstone
Here is a way to find them.
The Ultimate Public Campgrounds Project - Website Map
Here is a good source for RV park reviews.
RV Park Reviews - Trusted Reviews of Campgrounds & RV Parks
Your RV, at 37', is probably too big to use as a touring vehicle so plan to either tow or rent a car. If you rent you will need to plan your park stops accordingly and allow the additional time a rental takes.
Another caution. Unless you are staying at a commercial RV park (that looks like a parking lot) national forest service and the more natural campgrounds do not trim their trees to accommodate rigs. Expect to get scratches. Do not expect to get satellite TV reception if you want trees and shade.
If you do not have a Verizon cell phone I suggest you get one for the trip. Verizon has the best cell coverage in the mountain states. You can get one of the month to month ones at Walmart and just activate it for the month.
Many national parks have restrictions on dog such as:
Along established roads or in parking areas
In established campgrounds and picnic areas
They often aren't allowed on trails. However some places to allow them so you have to look at each park's web site. Preparing a matrix for all your stops with this type of info is very helpful when you have such a long trip.
While the big national parks are awesome to visit they are very crowded. Especially as Europe and other destinations have become less safe. Lots of international visitors. Maybe try to hit a few big ones and then seek out some of the more off the beaten tracks in between to get a variety of experiences.
I like some of the more low key experiences also. Such as:
You've got your pool, horse back riding, cook outs, rafting, stage coach ride for younger kids or pony ride, etc. This is like an ala-carte dude ranch. Our stay there:
1994 Brave 29RQ RV: Fort Robinson Nebraska State Park
Hope this is helpful.
Edit: Forgot to add that it can be very cold at night at altitude in June/July. If you have any plans that are at altitude make sure everyone has a coat.