I don’t see how anything he observed or was told changes the contract language - he was legitimately fired for cause. Trying to get more money from his contract is drilling a dry hole.
Extorting hush money from UT is not feasible (due to public disclosure) and extorting money from boosters is unlikely.
His best strategy is to keep quiet and try to rebuild his coaching career in the NFL. This is why I think “telling my side of the story” is just something to say, like “searching for the real killer” not an actual threat.
Edit: He might throw some dirt on Fulmer in his response to the NCAA, but If there is much to see there, it would have already been disclosed and Fulmer has been replaced.