Thanks for your replies. It's good to know that interns may still have a chance of getting hired later in the summer. The particular program that I'm going to be in is a district intern program, and I know that you must have a contract in order to participate. Since there is a six-week intensive training course starting in June, I think I will need a contract by then. The good news is that the job fair I will be attending is ONLY for interns in this program, so any schools interviewing there will be specifically looking for an intern.
Good luck and let us know how your job search goes!