Well, I went ahead and talked to principals in two other schools in the area. After meeting with them and some teachers, they were really into it and agreed to ask students in their extended (AP) courses.

This is fantastic and impressive. I hope you told those two principals what you think happened at your old school. You don't want that to come back and bite you. You will probably disagree with my following comments. I'm interested in knowing what you think:

Like all professional adults, she fucked off.

Not all. Some, maybe even many, deal with things with integrity.

But all of it could have been averted if only the mentioned principal would give me enough benefit of the doubt to actually let me explain it and talk it out like adults.

You are absolutely right about that. We should all give people the benefit of the doubt and try to understand a situation before taking action against them.

Principals are terrified of parents. Keep in mind that you still have not talked directly to the principal and don't really know what happened. You still have the option of behaving with integrity and trying to clarify any misunderstanding.

This means

1. Letting the principal know how you felt after talking to her.

2. Telling her what you heard and asking for clarification because

But one of the teachers told me why she thinks I'm no longer welcome.

is still just a theory.

3. Showing understanding for her concerns (if the teacher was right). Showing understanding that she is responsible for student well-being.

4. Showing your commitment to teaching, math, kids etc. by telling her that you needed clarification and you are seeking mentoring with regard to anything you do.

5. Hoping you hadn't caused any harm, and apologizing for whatever problem you might have caused.

Just because the principal is not taking you seriously, you can be serious with this false or unnecessary accusation and apologize and more importantly, point the way --

But that's not really a lesson for me: I know that most of the "adult and professional because of age" people are full of shit and more childish than I ever was.

Model appropriate behaviour by facing the problem calmly and asking for clarification and mentoring.

It's in your best interest to get this false or misguided or bullshit accusation cleared up with the principal and by all your actions, you've demonstrated that you are a serious teacher.

Rather than writing them off as being full of shit, give them a chance to work it out with you.