I've been out (by pure happenstance) for the last two full moons in Southern Utah, January in San Rafael Sweel, and last weekend at Bryce.
The best moonshots will be on the 16th, just before the full moon, when you still have some light on the landscape. The 17th and on the moon comes up too late, but you can do some nice long-exposure landscapes lit by the full moon after the 17th.
During winter, of course, the full moon comes up from the ENE direction.
I've always been parital to a moonrise shot with a long lens at Fisher Towers (about 20 miles NE of Moab), but in winter it might be a bit of a trick positioning yourself to get the Towers between the camera and the moonrise without the mesa behind the towers getting in the way.
The Windows section of Arches might be oriented nicely for a shot including Turret arch, and of course there is the obligatory balancing Rock with Moon shot.