I was there only once, but going back to look at my files it appears I was there on May 15, so my experience may be helpful for you. It was a very memorable trip for my wife and me and we plan to go back again. I think you'll like it.
The canyon winds around and there are galleries on either side so you won't have the same lighting at all of them. Some are located high up on the canyon walls and others are under overhangs so you have to be prepared to use different lenses to capture them. We reached the Great Gallery late in the morning and it was in full sun. A polarizing filter was helpful. The panels are quite large and are perhaps 20 feet above ground level so you will need a long lens to photograph them. One or two figures had shadows from the overhang, but most of the panels were fully lit. Cracks in the rock also cast small shadows.
Having been there only once I can't say what is "best." I got some good shots both of the galleries that were shaded and those in full sun. I don't think you would do well when the shadows from the canyon walls are cast across the figures--it would be worth waiting for the sun to move a bit to have even lighting on the whole figure.
The paintings are all somewhat faded so they present very low contrast subjects. Plan on some post processing to enhance contrast and color to see them to their best advantage in the final prints.
We left Moab very early to arrive at the canyon while it was still cool--just after sunrise while the canyon was still shaded. Nonetheless, when the sun reached the canyon floor it quickly became very hot. We did all our photography before lunch, had a sandwich, and hiked back out. We always take more water than we think we will need (and we have some experience hiking in the desert). This was one of the few times we wished we had brought even more water. The climb out of the canyon is in direct sun (at least in early afternoon) and we finished our water during the climb out; fortunately we had another bottle in a cooler in the car to refresh ourselves on the drive back to Moab.
It is a great location. Enjoy (but pay attention to the heat and dryness).
