No, you don't need reservations if you're flexible as to where you stay. You'll find a site.
The only exception would be Monument Valley. There's only one full hookup place and it Gouldings just outside the park. They are redoing the dry camping area in Monument itself so don't know when it will be open. That's our favorite - right in front of the Mittens. Otherwise you'll have to choose a park in a nearby city north of Monument but still, you won't need reservations at that time of year.
Ruby's is huge and there are many other opportunities in the area, including in Bryce itself. Page area has sites or great boondocking at Lone Rock, part of the national recreation area. You might want to check the Moab city website to find out if there will be a Jeep rally at that time but otherwise, there are plenty of RV parks and excellent dry camping areas in all directions. Dead Horse Point State Park is excellent. Have fun!