The problem is that it is a little late to be planning an award stay at high demand properties during high demand periods, unless one has a bit of flexibilty. Award nights for Boscolo Exedra are available on the following nights if it is of any help:
June 3, 16, 19, 21, 22, 26, July 2, 3, 9, 10, 20, 21, 25, 28, 31, Aug 2-4, 6, 8-14, 16-Sep 3
In June and July there are a smattering of dates, and August is wide open with the exception of 3 nights.
To answer your question: No, awards generally don't open up as you get closer to your stay. Quite the contrary, available dates for standard awards disappear as Classic guest rooms are sold out. Classic guest rooms appear to be Boscolo Exedra's version of a "standard" room. The only way an award would open up is if someone were to cancel a Classic guest room reservation.
I hope that helps.