The union owes its members a duty of "fair representation." The duty of fair representation differs from the duty a lawyer owes his client: The lawyer owes the client a duty of undivided loyalty; the duty of the union toward the member is always divided, because the union has to look out for the whole, and the individual member may get shafted if the union decides it's in the interest of the whole to overlook something the individual member is complaining about. They can't treat you arbitrarily, or refuse to process your grievance, but the nature of the duty owed is a divided one, because it's the union that bargains with management to reach an agreement which binds ALL union members, not just you.
I'm not licensed to practice law in Florida so don't take what I say here as legal advice. It's simply my analysis of the facts you present in light of general principles of law, for the purpose of educating. If you need legal advice, please consult a lawyer who holds Florida licensure.