It is difficult to be objective about ourselves, especially under embarrassing circumstances, but if what you say is true and accurate, it is possible that the school administration sided with the professor unfairly.
I was once shocked to be falsely accused of 'helping' a fellow student when I simply confirmed the question (NOT the answer) in an official community college online forum for a hybrid class. The question I answered resulted from a delay in receiving the textbook and the student was afraid he had jotted down the wrong question from the library. I never even hinted at an answer but was rebuked by the professor. This was early in the class and I don't recall any serious ramifications (incidentally, this teacher turned out to be so completely incompetent in the class subject matter that I chose to meet with the department head (CIS) and academic dean to discuss her 'teaching' style. She is now retired.) My point is that faculty are not omniscient nor infallible.
From professional experience, I have found it works out best to be up front and honest as soon as possible rather than leave things in a somewhat murky grey zone for extended periods. This is easiest if you have a sympathetic professor or counselor to help guide you thru any appeal process and are willing to do what is necessary to right the inadvertent wrong rather than justify the wrong doing however innocent.
From what you describe, you seem to have had little or no motive for posting what you did other than intellectual curiosity, a trait that should be vigorously encouraged in education. Assuming this to be the case, it still does not speak well of your judgement to presume that you could post something academic online and no one else would find it. Search engines are remarkably capable tools as you have discovered. Even if not mentioned, you must assume professors use plagiarism detection tools etc. Putting the best light on this rather big negative, one could argue that posting publicly is further evidence of your innocent intentions.
Given the apparent grey zone into which you fell, perhaps you could collect any policies on academic dishonesty published by your school as well as the class syllabus given out by your professor. Read them and summarize how well you complied with them (if you did).
The fact that other professors did not give you an 'F' for similar previous posting is not any more convincing than telling a police officer that you have been speeding in that section of town every day for a year before and never got a ticket. The fact that you have been violating the rules regularly might only add to your sentence.
However, if you feel singled out unfairly try to find similar precedent for online posting from yourself or other students that was endorsed or approved by the professor. Try to find an unbiased person (not family or close friend) or better yet, professor that you respect willing to review the particulars of your case and give you advice. Appeal any adverse decisions as appropriate. No truly innocent person willingly accepts guilt.
Most importantly, learn from this experience and become better, not bitter. Keep in mind the big picture: Academic dishonest is unethical not arbitrarily, but because it deprives you (and other students) of a full education and honest and fair evaluation of your accomplishments.