12R34Y,
Wow, you're in a tight spot. It sounds like you've already sent several email reminders to your 2 profs, but they aren't responding.
Did you give them a deadline? Some profs need hard deadlines as much as some students do
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Did you give the profs addressed envelopes so all they'd have to do is write the letter and plop it in? I've found that to be effective. If you haven't already done that, that might serve as an excuse to stop by their offices--that is, to drop off addressed envelopes and to remind them that the deadline is very soon.
If you've done all these things already, I'm not sure what to say. I'm assuming these profs were enthusiastic about writing a letter for you. If not, that might be another concern.
I think setting a firm deadline might be your best bet--and not a deadline that THEY come up with. You have to make it a deadline that counts. Emphasize the external consequences.
Unfortunately, profs are occupied with all sorts of stuff in their own lives, careers, and with hundreds of students. Some are better at handling it than others. I'm sorry you have such duds in this area.
I would actually need more details to provide better advice. On the one hand you say you're not getting repsonses, on the other hand you said they claim it'll be "next week." Could you clarify? When were you getting responses? How many emails have you already sent? My husband's a prof by the way, and I'm trying to best guage the kind of situation and personality you're up against.
Are you in a position to ask some other profs for a letter?
As for good ways to get letters:
1) make sure the person knows you well and is enthusiastic to write a good letter for you
2) provide a written summary of info about you that might help the prof formulate the letter
3) provide addressed, stamped envelopes for ease
4) provide a hard deadline (and reasonable time). I usually make the deadline what I want it to be, and much earlier than the real application's deadline.
5) if writing a reminder, which some profs like by the way (they know they forget these things), you can start it something like:
Dear so-n-so, last January you generously agreed to write a l.o.r. for me. The deadline is X, only one week away, and I just wanted to ensure that your letter would be in on time. Thank you...etc., blah blah, sign your name.
You get the idea.
6) As someone previously posted, I also would stop by their offices just to chat, and also to check on the status of the letter.
best of luck,
--kris