Sorry! I'm not so convinced it's a good idea either. June is one of the top, top months for travel not only in the U.S. but also from the U.S. to almost anywhere on the globe.

In reference to your ticket price mention. In my humble opinion a destination's ticket price is in direct correlation to your chances of actually making the flight. The higher the ticket price, the lower your chances. The lower the ticket price, the greater your chances.

Best of luck! Take a backup pass! Keep your options open (i.e. if you're trying to get to Rome then don't turn down their offer of a flight to Milan, for example).