Around Moab: Dead Horse Point State Park, Corona Arch, Fisher Towers, Castle Valley, Behind the Rocks Wilderness Area, Kane Creek Road, Potash Road, Hwy. 128, La Sal Mountains incl. the Loop Road.
Not far from Moab: Goblin Valley State Park, San Rafael Swell, Grand Staircase-Escalante, Monument Valley, Goosenecks State Park, Cedar Mesa, Kodachrome Basin State Park.
Trust me, there is no shortage of spectacular locations outside of the national parks in southern Utah. I'm leading a private workshop next week with a couple clients and we're spending two full days together, photographing at least 8 locations, and not one of them is within the boundaries of a national park.
Good luck and have fun!