Hey Helix, this happened to me last year -- actually, one of my recommenders totally missed the December 1 deadline. Afterwards, I called the school pretty much in a panic, and they said that they definitely accepted (somewhat) late letters. When I finally got ahold of the prof, he told me the same thing -- that the deadlines are mostly for applicants, not recommenders. He submitted the letter later that week, and I eventually got an offer from that school, so all was well.
I guess what I would recommend is for you to call the school(s) that you are concerned about, and nicely ask about their policies on late letters. If they absolutely don't take late letters, then I would hunt that professor down, since he/she may be operating under the assumption that my prof had (that the deadline is for the student, not the recommender). They might say that they have a little leeway on accepting late letters, however, and in that case, you wouldn't have to feel so panicky right now.
And, if you find out that the deadline is VERY firm, I don't think that you need to worry about being obnoxious. BE obnoxious, if necessary.