I was on academic probation
Well then you weren't dismissed for failing a shelf exam; you were on academic probation presumably for struggling elsewhere along the way, and this shelf was the straw that broke the camel's back. Not trying to be mean, but trying to play up the injustice as being greater than it was ("It was just ONE shelf exam") does not make it seem like you are taking responsibility for the situation and won't look good as you try and make another appeal.
I won't mince words: that sucks, and I'm really sorry this happened to you

As others have said, really your only recourse would be if you could argue that they didn't give you due process via the appeals process, or that they wrongfully denied you accommodations for a medical condition (I'm assuming ADD); without knowing more about your school, its policies, and how they handled your specific situation, I'm not sure where you'll get on any of those arguments, but if you really want to go down that route I'd suggest meeting with a lawyer to find out if you might have a case on any of those grounds. The alternative would be if you have faculty members that you have an excellent relationship with who can talk about your ability and would be willing to advocate for you that the committee reverse their decision.
If all of that fails, I would echo that you could look into a PA program. It will be a tough pill to swallow, but I have friends who were dismissed from med school and eventually became very happy with PA school.
Good luck.