I've been away for a while, gone to Austria for easter w/e skiing...and then didn't check immediately
the threads, sorry for not answering immediately.
Now, as for your question:
The winter season ends in the Dolomiti area in these weeks (looks like 14.04.2002 will be the last day),
so until mid june it is kind of a "dead season" as far as Italians, then mid-june will start the summer season...
As you too well know, weather in the mountains is unpredictable but, may-june are transitions months, weatherwise
too. Given that this winter has been a "little bit dry", I'm fearing murphy's law, and I am expecting a "wet" summer...or at least a "wet" springtime...
In June the temperatures in the mountains are cool (not cold), but when the sun shines it's very agreeable, I've been there (not in Cortina, but in the area) in summer only a couple of times, and in August, traditionally the hottest summer month, and weather was superb.
Since we are talking about Cortina, thus the area "big town", I am confident you'll be
able to find a room for two for the night.
I took the liberty to ring up the ATP Cortina (Tourist Board) and there they told me that
surely there will be no problem as far as a free room, they just do not have an up-to-date list of the hotels that will be open during the period of your visit,
and to call them in a couple of weeks...then they should be able to tell me how many hotels will be opened at the end of may.