Matt, Perry is right, but you may have an option. Last year our Moab run was the weekend before Memorial Day (not after as this year's is). My son and I returned via the Ouray area and arrived the Tuesday before Memorial Day. Ouray is heavily dependent on 4wd tourism and tries to open the Alpine Loop pass by Memorial Day. No one in town knew how far the trail had been cleared though; not the city nor the two Jeep rental shops in town. The only way to find out was to attempt the trail ourselves the next day and find out.
Wednesday we tried to take the Alpine Loop. We made it up the Engineer Pass trail to Animas Forks ghost town and played there awhile. Continuing further, we went another mile or so up before catching the bulldozer plow and workcrew that was clearing the trail. Snow was a good two stories high there. They were hoping to clear the trail to the summit by the holiday weekend and meet another crew working the other side at the top.
I returned back to Animas Forks and tried Cinnamon Pass which was lower in altitude. It was cleared, so we made it all the way across, but it took a full day trip to finish.
Because Ouray makes such an effort to open the pass by Memorial Day, I believe you have an excellent chance at enjoying the full Alpine Loop after our Moab run this year, but go prepared in case it hasn't been fully opened.
That is the most famous trail in the area and will probably be the only one open, but it is awesome and worth the trip by itself. Poughkeepsie Gulch entrance off of Engineer Pass was 12-15 deep under snow and the same could be said for entrances to Black Bear Pass and others. I had run Black Bear the year before and knew exactly where the entrance was and still couldn't find it since all signs and landmarks at the entrance were under snow two stories high. If you've never been on the Alpine Loop and feel like driving over there for that one trail, I say go for it. You won't be disappointed.