If you have experience with your winter climbing gear, have done Rainier in May, and have been on the Whitney trail in June, I'd think you won't encounter anything that you can't handle. The accidents mostly occur on "the chute" (actually the snow slope west of the 99 switchbacks below Trail Crest. That's where people with less than optimum gear and experience (ice axe and crampons) may lose their footing and slide and get hurt.
I've not heard of any snow accidents between Trail Crest and the summit. What you're thinking are shady areas (that I can think of) are the west-facing slopes. They melt out before the chute, and shouldn't give you any trouble if you are using the proper gear.
