Recommender Too Busy to Submit Letter
It's hard for me to understand how this can really be an issue for the professor, because in this day and age of online submission, once she has written a letter, it takes very little time to submit it to each additional school.  Twelve schools is a little on the high side but certainly not ridiculous.  However, it won't really help the situation for you to tell her that.  Of course, overwork and burnout are definitely things for faculty, and there could be issues like that in play.  But my point is that based on what you've said, it sounds like your request is totally reasonable, and this is unprofessional behavior on her part.

I would suggest you discuss this with her department chair.  You could share your email chain with the chair, making it clear that  you requested the letters well in advance and the professor agreed to send them; that she has actually already written the letter; and that the reason for her balking is not apparently related to any concerns about your qualifications.

The chair will be better positioned to speak to the faculty member about sending the letters.  Departments have an interest in seeing students succeed, and I think most chairs would not be happy about seeing this obstructed for relatively trivial reasons.  The chair may also be able to offer to adjust the professor's workload, or have the letters submitted by some other staff member.  As a last resort, the chair may be able to lean on some other professor to write a substitute letter at short notice.

Be prompt about this, but don't panic too much about approaching deadlines.  Graduate programs are usually able to have some flexibility about letters submitted after a deadline, as long as the student's portion of the application was in on time.

This might hurt your relationship with the professor somewhat.  However, I would say she already blew up your relationship by refusing to submit your letters.  That's a breach of a very straightforward professional obligation.