A lot of the West is public land, ( Utah for example is 74% public land) hence you won't find a lot of campground that cater to horses like you do back where you live. We just pull off into the trees and do what is called dispearsed camping. True there are no hook ups, it's all primative camping. But it's free.
Since the land is managed by government agencies ( Forest Service, BLM, State trust lands etc) you have to abide by their rules, one of which is certified weed free feed.
Do your research on what areas you want to ride. Don't worry too much about dude ranches. They focus more on people with out their own horses. Check out places to stop along the way. I usually pulled into fairgrounds and turned the horses out in their corrals for a stretch.
Once in the mountains, I usually just high line my horses between trees.
Parked in the trees off to the side of road.
Here in Utah, A lot of the popular trail heads have corrals built by Back Country Horsemen.
I'd suggest lots of places in Utah and Western Wyoming, But since you are not coming that far west. A loop through Western Colordo and up into Wyoming will give you a great taste of the west.
Also research what time of year you are coming. There can be serious snow pack still in late May or Early June in the high country. We are usually still riding the high desert country of Southern Utah in May because the mountains of Utah are still snow bound. But rides into Utahs San Rafael or Bryce Canyon in May can be breath taking and much more comfortable temperature range.
Also a lot of trail heads in the west service one or two main trails. They are mainly designed as jumping off points for extended trips. If you are going to return to your trailer nightly vs packing your camp with you, you will repeat a lot of the base trail, Hence I recommend that you be comfortable moving to another campe every couple of days. You don't have to move far, Often a 20-60 mile drive, will put you into a completely different area to ride.