I've usually found that the best images are found somewhere on the way to or from the "iconic" sites.
Now you've done it. You've told everyone the secret.
I would like to see some of the iconic sites, but I am not interested in photographing them in other people's tripod holes. I tend to like to the out of the way places and unusual details. Plus I find that shoot and go does not work well for me. I need to know an area to see well.
I think Cacho might be right up yer alley. Its relatively out of the way but easy to get to, unless it has rained. Mesa Verde is spectacular, but geared to a shoot-n-go public.
I recommend taking at least a day to explore the area around Santa Fe and Taos. Daniel Gibson's "Pueblos of the Rio Grande, a Visitor's Guide" is quite useful. If you have only one day, you can't go wrong taking the high road to Taos, visiting the Santuario de Chimayo (and Rancho de Chimayo for lunch), San Jose de Gracia in Las Trampas, San Francisco de Assis in Taos, and Taos Pueblo. Each of these sites is worth seeing regardless of photography, and each is compact enough that you can walk around, contemplate, AND make an image.
For a more out of the way experience, requiring 4x4 and hiking, the Coomb Ridge/Cedar Mesa area is wonderful.
Canyon de Chelly has got to be on the list. The Thunderbird has always been reliable and there is a newer Holiday Inn close to the entrance. You can walk in at White House Ruin and walk around a lot at Spider Rock overlook. (Be sure and check out the caves off the rim at Spider Rock.) The canyon tours are interesting but they don't pause for photographs. Guides are 30.00 an hour for the canyon bottom. Worth it. They have jeeps of their own or you can take a 4WD in. You can't get into ruins or approach them closely but plenty to photograph. Like being inside a sculpture.
Shiprock has no ruins that I have seen except a couple of very little rockwork places that are of unknown origin. Might be some kind of spirit/quest site. You do see prayer wheels around it, some made with beer bottles, and those little burned corn altars. Two graves up on the rock, one probably only holding ashes but the other seems to have a whole person in it. They are in obscure spots and easy to overlook. Shiprock quite beautiful in whole and in detail. Keep your car in sight. There isn't any local security that would be interested in you. Closest place is to stay in Farmington- 40 miles one way. I usually truck camp but you have to be careful there. At the popular spots the party could start at 1:00AM.
Chaco is worth a visit. Campsites only. If you need it, bring it.
To mention two Anasazi sites not yet mentioned, Keet Seel and Betatakin are in the same Navajo area you are contemplating. For both, the hiking season begins end-May, which might be too late, since you mentioned mid-May. Keet Seel might be the best-preserved Anasazi site open to the public, because it is a hard 8-mile hike each way, but it is something of a life-time experience. I did it years ago, camping overnight in the NPS camp site. You can only tour the site with a guide, and cannot set up a tripod within the ruin, but you can photograph from the camp site and into the ruin.
Along with many others who have posted, I would consider Chaco Canyon a "can't miss" especially because the photography is unrestricted and the ruins are extensive and wonderful. We camped there (suspect that is really the only way to do it) and we were visited by both coyotes and the ghosts of the Anasazi. You won't forget the experience.
Hovenweep is one of my top sights too. I was there on a BLM Visiting Artist gig last spring and according to the rangers the camping spots never fill.
The choice of easy-to-defend, out-of-the-way dwelling sites, especially here, convinces me the Anasazi were often under siege – or warring w/ each other.