It sounds like you have an overly idealistic view of how this process works.

Why do you imagine non-disclosure agreements are standard after an out-of-court settlement? It's because "pay me or I'll badmouth you" is essentially always one of the implied threats.

You can call it blackmail if you want (and I'm being deliberately glib here, because I also think it's at least blackmail-esque), but your lawyer and the company's lawyer will absolutely understand that this is how things are done.