There are lots of dispersed camping opportunities and smaller public campgrounds that do not take reservations in the Four Cornets area if that style of camping would work for you. For Lake Powell and Glen Canyon NRA, check out Lone Rock Beach primitive camping and Lee's Ferry. We stayed at and enjoyed both when in that area last Fall. Kaibab NF and BLM lands also offer a number of possibilities so I don't think reservations are a must unless you really want to stay within a certain Park.
We'll be in southern Utah for a month next May with more than a dozen other campers and have made reservations only for Watchman and the North Rim. The rest of the time we have a primary spot picked out along with two or three nearby alternates if the first choice is full.
Hope you have a great trip.
