Are you crazy for wanting (getting) a new job late June/early July, nah.
Will it be a perfect situation (if you move) for your current school, nah. But life will go on.
I don't think that asking during the interview what things are currently in place is out of line (if admin even knows). Like you said if they is NOTHING in place then maybe you don't take the job. But better to have all the information you possibly can so you can make the best decision.
People get hired "late" all the time. In my current job my first day was August 5 with school starting August 22. Luckily a show and drill was in place.