I live in Moab.
First, April/May are two of the busiest months in Moab and the surrounding areas.
There are quite a few private campgrounds in and around Moab; basically on Hwy 191, the major (only) north/south hwy into and through Moab. Reservations during April/May could be hard, but probably not impossible to get. Plan as early as you can to make reservations. These are probably the ONLY locations where you will find shade trees.
The area is surrounded by almost 25 BLM campgrounds but they have NO services; other than after mid-April, the two campgrounds 30 miles south of Moab have water. They are also on a first come/first serve basis, and best time to grab a site is on Monday afternoon or Tuesday. They will be very busy. Not many of the sites will have any noticeable sized shade trees since this is considered a high-plateau desert.
BLM campgrounds on Hwy 128 (known locally as the river road) that runs along the Colorado River has some nice campgrounds but they are very busy, and not can take a long trailer. North of town, off Hwy 313 - the road to Canyonlands Island in the Sky and Deadhorse Point state park, has a large campground called Horsethief, it can take long trailers in many locations, tends to be a bit quieter, but again, not many large trees. (I won't even mention the campgrounds along Hwy 279 (Potash Road) or Kane Creek. These tend to be small campgrounds or extremely noisy due to ATVs/UTVs and motorbikes.)
There are multiple areas where you can legally disperse camp on BLM and/or lands owned by the Utah Schools (hard to explain but yes, there are trust lands set up for schools). Needless to say, no dispersed camping in National Parks.
North of town, off Willow Springs is an area noted for dispersed camping on the school lands. Also, off Hwy 313 there is a dirt road, Dubinky Well, that has a lot of areas where you can set up and camp for free. Usually a lot quieter up there than on the River road, but can be dusty.
For more privacy, and more shade, you will have to go up into the Lasal Mountains, south of Moab. I am not sure about hookups, but there are two campgrounds off the Lasal Loop road; Oowah Lake, and Warner Lake. They are run by the US Forest service.
Week before Eastern Sunday is known as Jeep Safari week and there are, on average, thousands of Jeeps/FJs/Hummers in town doing off-road trails from scenic dirt to extreme roads that usually end up breaking parts.
It is also the start of the busy SAR season for Grand County. Those choppers overhead are either tours. life flights to Grand Junction or SAR.
There are two national parks outside of Moab. Arches, about five miles north of town, has a campground at Devil's garden but it is usually booked at the start of the year. I do believe they have some on a first come/first serve basis, but not positive. And I don't believe there is any hookups, just water.
Canyonlands, Island in the Sky District, about 35 miles northwest of town, has a campground but not sure about hookups and if like Arches, always booked.
There is Deadhorse State Park campground, also about 35 miles northwest of Moab, but there is NO privacy in a Utah state park campground.
To sum it up, not much privacy during these months unless you disperse camp. Not much in the way of shade from trees except in private campgrounds.
However, it is a great place to visit, very scenic, lots to do like the Nat. Parks and hiking trails galore.
List of private campgrounds:
BLM campgrounds (a few missing on this list):
Dispersed camping areas and restrictions (if any):