I'm very sorry about what you're going through. The key to the answer I'm about to offer to your question is the following general observation: the people in academia are not robots. Despite how it looks from the outside, all these rigid rules and deadlines that we have were made by humans and were designed to serve specific purposes. When we judge that the purposes are not served by these rules, it is often possible for us to ignore or bend them, within certain constraints of law and common sense.
With that said, here's a way I suggest to handle the situation that will leave you with a fighting chance of having your grad school applications be considered seriously in this application round. The most urgent thing you should do is write an email ASAP to the programs you are applying to, in which you explain that you were involved in a serious and irredeemable confrontation with one of the professors writing your LORs. You don't have to explain the precise nature of the confrontation (I don't have a strong feeling about whether it will be to your advantage to explain it, and there are obvious privacy and even legal issues that may make it problematic to share too many details), but either explain it, or if not, do your best to stress how severe the situation is and make it clear that you are a victim, not an instigator, of the conflict. I.e., find a way to make sure they understand that "something really bad happened, and it wasn't my fault" and sound sincere without providing details that are too sensitive for you to want to mention.
Then explain that as a result, your LOR from that person (assuming it is even sent) has become invalid, and say that you would like to make two unusual requests, which are, first, that they give you a deadline extension to allow you to obtain another reference, and second, that they discard any LOR that may be sent by the professor with whom you are in conflict without looking at it.
In the email, emphasize that you are only asking for an exception to the rules due to the extreme and unavoidable nature of the situation, and emphasize that you will still submit your application and all other materials on time by the original deadline, so that the amount of rule-bending you are asking for is as absolutely minimal as possible.
In parallel to sending the email, start looking for an alternative letter writer ASAP. Ideally it would be someone who you think would be sympathetic to your situation and whom you would feel comfortable confiding in (I imagine a female professor might be more likely to answer that description, for obvious reasons). It may be that you would find someone sufficiently understanding that they would be willing and able to write the letter on short notice, as others have suggested, which would allow you to even meet the original application deadline; I certainly can imagine myself helping you in precisely such a way in a situation like this.
Finally, in this answer I wrote to a somewhat analogous question I also suggested trying to find a local ally such as the department chair who could also contact the places you're applying to and advocate on your behalf, since you are making unusual requests which the places you are writing to may feel inclined to ignore. That situation involved something much less serious than a sexual assault, but the advice may still apply (though in your situation a tricky bit is the problem of how much information you're willing to share; if you do not want to talk about the sexual assault that makes things potentially more difficult).
Good luck! I can't be certain that what I'm suggesting will help, but I don't think you're asking for all that much given the circumstances, and think there's a reasonably good chance the people you write to will be accommodating to your requests. One last remark is that I don't think it makes sense to ask the departments to waive the requirement of three letters and ask if they would allow you to submit just two. The problem is that this would make it impossible to compare you with the other applicants on a level playing field, which would create an ethical and perhaps legal dilemma for the admissions committee, with the likely result that they would be forced to reject your application even if they may be sympathetic to your problem and wish to help. But I'm not completely sure about this point - if anyone else can offer more authoritative advice on this issue that says something else then I'm open to being convinced.