I just noticed that you will be driving. Correct? Convenient, but you will miss the spectacular Bernina Express train ride.

Not sure about your price range, so I give you two recommendations on accommodation (from personal experience):
*** Hotel Waldhaus am See, St. Moritz
(www.waldhaus-am-see.ch, I am afraid German only)
***** Hotel Waldhaus/Sils (www.waldhaus-sils.ch)

Both are excellent and should be open from mid June on.

Ingo