I think all the southwestern national parks are already plenty busy by May. I visit them in April-May because it's too hot after that. I've been at the NPs on a cold blustery April weekend with snow flurries blowing around, and had busloads of pushy French or Japanese tourists mow me over. But it stands to reason that they get worse after the kids are out of school. So I guess you could say, no matter how bad it is, it could be worse.
As a general rule, I don't make reservations in advance when I'm on a trip. And I always find a place to stay. But around the NPs it might be different because of the crowds. You can always find a bed if you head down the road a little ways. Anywhere near a NP, I wouldn't wait until almost dark to start looking. Whenever I plan on staying in Moab I get a reservation just to be safe. That's partly because Moab is kinda spendy and the cheap little mom-n-pop joint I like only has 8 rooms. I think you would probably be safe if you don't care how much it costs.
Remember that 99% of the visitors in southern UT are going to the NPs. I like to say we keep the parks around just to keep the tourists out of our way. LOL. The beauty of southern UT doesn't end at the park boundary. In between the parks you'll find gorgeous country and little towns with peace and quiet and plenty of room. Places like Kanab. Escalante. Mexican Hat.
One other thing. I don't want to be preachy. But don't let the long ride from AR to the west get you down. A lot of people think that the best way to cross the plains is to get on the interstate and put there head down and grit their teeth until they get past it. If you slow down and take the little roads you might actually enjoy it.