What happens if no one wants to advise me? I was accepted to a PhD program. I am finished with quals and looking for an advisor. I'm having a difficult time finding one though. Everyone is not accepting students (or at least not me). It's a small department. I'm worried I won't find ANYONE to advise me. I finished my quals, I work hard, I try to be polite.

If no one wants to advise me, do I just have to drop out?
You have to have an advisor to continue in the program, and advising needs to be a mutually consensual relationship.  So if it really happens that absolutely nobody will advise you, then you'll need to leave the program (but you could probably transfer to another one).

But in practice, it should very rarely get to that point.  Your program should have a graduate chair or some similarly-titled faculty member who is in charge of the PhD program.  Talk to this person.  There are a couple of possibilities:


There are faculty who would be interested in working with you in principle, but they feel too busy.  The graduate chair can try to convince them, and exert pressure.  If the department is interested in having a good PhD program at all, they won't want to lose good students for no good reason; so eventually someone should be willing to take you on.  Maybe other workload bargaining will happen behind the scenes (e.g. Professor Smith says "I'll advise the student, but I want to be excused from serving on Committee A.")
There is something about your academic background or personal style that makes faculty reluctant to work with you.  This is less likely; if there were serious issues you probably wouldn't have made it this far in the program.  (In particular, you've passed quals, which is a very good sign.)  But if there is some issue like that, the graduate chair should be able to talk to you about it honestly; and then you can work on addressing it.
Least likely possibility: the program is totally dysfunctional, and faculty members don't really want to invest any effort in working with grad students, or are so completely overwhelmed with other duties that they cannot.  Hopefully this is not the case, but if after talking with enough people (including those at other institutions) you become sure that this is what is going on, then you should transfer to another program as soon as you can.