How can I politely remind my professors about a recommendation letter they promised? I am currently applying to a masters program, application is due on April 15, recommendation letters can be given until the 23.

I asked 3 professors to give me recommendation letters approximately on march 14.
One of them already submitted it, but 2 haven't.

I don't want to be excessively insistent, because A) they did tell me they were busy when I asked, and B) I made a grave personal mistake towards one of the professors a long time ago, and the simple fact that he agreed to give me a recommendation letter is, I think, enough for me to never ask him for any favors in my life time.

However the due lines are coming, and I am getting a bit anxious.
Dear Professor X,
  
  Just a friendly reminder that my recommendation letters to schools A, B, C are due on [dates].
  
  Thanks very much for writing letters for me.  Please let me know if any questions come up.
  
  Sincerely,
  
  Your Name


It's as simple as that.  No need to overthink this.

Note that it's pretty normal for professors to wait until close to the deadline to write recommendation letters, and shouldn't in itself be cause for concern.  Part of it is the natural tendency of people to put things off until they are urgent.  In the case of recommendation letters, there are also a couple of rational reasons for it.  One is that in case something significant happened to you between March 14 and now,  either good or bad (published a paper, received an award, failed a class), they'll be able to mention it in their letter.  Another is that if you had changed your mind about applying to some or all of your schools, they wouldn't have wasted the time writing and submitting an unnecessary letter.

Anyway, you don't get to be a professor without having a lot of practice in successfully meeting deadlines, one way or another.  It is unlikely that they've forgotten about your letter, but still, a reminder doesn't hurt.