You'll be arriving during one of the busiest holiday weekends of the year (Memorial Day). Many people will have exactly the same idea as you, and the proprietors will charge increased rates because they can.
However, if you are very flexible and don't mind camping and don't mind moving around a little bit, you can simply drive in any random direction and you'll have many options. You may have better luck finding some place for a week-long stay starting the following weekend (June 4), when all the vacationers are back home. But if you plan to do this on such short term, you will need to get a little lucky to find a place by just randomly showing up. This is the high season. Be ready to pay steep rates.
I assume you're coming through Utah to Las Vegas. You can't go wrong there either. Point the car in any direction and you won't be disappointed. However, it may be getting hot the lower elevation you go. I love the Moab area.
