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Sending in your app without it's LOR is basically pointless. The won't look at it. You should ask your professors to revise that deadline back to November/Thanksgiving

"Prof. Jones, I really apologize, but I didn't realize how important it was to get law school applications in early. I know I said December, but if you could get my letter done by early November, I would really appreciate it."

It's a 10-minute job for them to write a LOR. You should just have to communicate that your deadline has moved and if they're decent humans, they'll just do it.

^This. I mean, it's the beginning of October; usually I think it's polite to give 3-4 weeks notice for writing a LOR (I've had students give me way less). So just ask if they can submit it earlier. (I do think if you give someone a deadline - for anything - you have to expect they'll do it right before the deadline. Some people are way on the ball and don't do that, but most will.) For one thing, semesters are less busy now than they will be going into December, so it's really better for your profs this way.