My dogs favorite desert hikes all have one common theme. Water.
1.) Negro bill canyon is a tributary of the colorado river 3 miles northeast of moab, hosts a perennial stream and the impressive morning glory bridge.
2.) mill creek canyon, located just east of Moab. Is a gateway zone to the la sal mountains which harbors many natural delights.
More ideas...Pritchett or hunters canyons, Klondike bluffs, fisher towers, corona Arch(entire Trail is on slickrock), conehead valley, canyon rims recreation area(aka poormans canyonlands n.p.) and just outside needles district indian creek, davis and lavender canyons.