Dead horse Point is a Utah State Park and it definitely fills quickly. The other BLM campgrounds can fill quickly it really depends on where in Utah. IE the ones around Moab certainly fill quicker than the ones out in the San Rafael Swell. One of the nice things about BLM is while they have campgrounds they also allow a lot of dispersed camping regardless of campground or not though there are areas that are closed for camping best to check with the local office in the area you plan on going too.
Utah is becoming if not already a victim of its own beauty. 30 years ago I could go out there and not see anyone for days in the back country if at all or get a hotel in Moab or any other small town, camping spot no problem day of arrival. Now you darn near need to plan 6 months in advance and the place is crawling with people from all over the world. Doesn't keep me away but I do long for the Utah of a few decades ago, undiscovered by most. I certainly understand why people want to go out there and experience it as I do. I just wish the masses would care for the place. Litter, dog poo on trails, dogs in prehistoric ruins, vandalism, disregard for the locals, and a seemingly total disregard for proper etiquette for other people. While the previous statement obviously doesn't apply to all people or dogs for that matter it applies to enough of them that it is causing the de-beautification (I seriously doubt that is even a word) of the place. Sorry for the rant there you asked a simple question and got my worthless words instead.