Can you let the second school that while you know they are very busy with their process, you want them to know you have a job offer, but would much prefer to work with them. Ask them if they can help in anyway?
If they can’t help, then you’ll need to figure out if you want to go with the sure-thing or the job you’d really like. Do you have any gut feelings about the 2nd job? I think if your 2nd choice is quite interested in you—they’d work with you.
A third alternative would be to accept the first job, and if you get the second one quit. To some this would be unprofessional. And if you are in a system where all schools talk to each other—this may not be the best thing to do.
However, you have to look out for you—and this happens quite a bit (folks quitting after just starting a job). The good thing too—is the 1st job still has a bunch of applicants and resumes; they’ll probably just go on to their 2nd choice. And in truth, when you accept job #1, you have no concrete idea that #2 was going to hire you. By the time #2 hires you, company #1 won’t have invested too much time and money in you.
Most people understand that one would quit for a job they like better. The company may not be thrilled, but they should understand. And remember, you cannot control how company #1 takes your departure. But really, you just started, and are not irreplaceable.
Just be sure #2 is a sure thing—like having something in writing before telling #1.
Bytheby, I've done this more than once--and I'm still okay...
Good luck!