I would find e-mail addresses and phone numbers and see what you can do. Did you make a copy for yourself or just mail the one-and-only? I would not worry too much about the copy if that is the case, I imagine they will make a copy, they often have to anyway so they can give it to various people looking at it, etc. I think the problem of incomplete applications is probably pretty common for them and the "due date" is just for that reason, so they have time to get things "together" for your file and then start looking at things. Otherwise people would never reread the instructions like you did and would be forever missing pieces here and there and it would be harder for them than it is.

If you are graduating, I would look into some day going to graduate school there if you do not get into this program? Dreams usually don't have time limits, it's possible to make the important portions of them come true at just about any time. I'm doing one of those educational ones now and I'm 63