Congrats! Is that the usual time of year when you go to FW? Do you typically book this close to your vacation date, and has there ever been a problem finding a site? We typically book our resort stay 6-8 months in advance, but that's because we have some kind of promotion (e.g., Free Dining) we are trying to take advantage of...
June is usually not busy. If you are booking for a week you should have no trouble getting in.
We take our bikes to FW and ride the loops and it is surprising how empty the loops are. If you put eveyone together there would be easy 3 or 4 empty loops.
There is never a bad time to do Disney. We grab a morning riding our bikes there whenever we can.
Enjoy