Beating the crowds may be easier at Wild Iris, but only because there are fewer established climbs compared to Ten Sleep. As for snow, the road has been cleared but there is still snow along the road and in the trees. There isn't much snow along the main wall of the Wild Iris. No clue about OK Corral and campsite areas.
After many years of going to Wild Iris, I finally went to Ten Sleep recently. It's like 10 Wild Irises. They are both worthy destinations. You can't go wrong and there are enough climbs at either place, that crowds wouldn't be a deal break. The climbs at Ten Sleep are longer in general. There is also great fly fishing in Ten Sleep Canyon if that's a consideration.