This happened to me. Go in person to the office if it's at all possible. Tell them you understand that they/he/she/whoever is extremely busy and dealing with your rec is a very small part of their job. However, you are concerned that your applications will be incomplete at a very near date and will therefore not be processed. I think a personal visit tends to speed things along...very rapidly. This is an important deal and there is no reason why your adivising office should not have that letter done by now. I went over the person sending the letters in the office at my undergrad and directly to the person who wrote them (sender person told me it was not written, which was not true). Do what you need to get the letters there, if you asked a long time ago there's no reason for them to be angry about you wanting things there on time.
