Private v. public doesn't matter. This idea that the government can't be sued is crazy. Universities get sued all the time and pay up all the time.
They do have access to sovereign immunity, but that defense is typically waived to the extent there is insurance. I specialized in municipal law back in the 80s and early 90s so I know a lot about this area of law. Typically states have some type of what is called slot insurance. The idea is that the "state" should be on tap for damages. If their driver hits you, you should be able to recover. If they violate a contract, you should be able to recover. But, it makes for a juicy target. So many jurisdictions will insure up to, say, 1 million dollars, but then not up to 5 million dollars, then anything over 5 million. The balance/give and take is that we want to protect our citizens from the wrongdoing of the state, and a million in insurance does that for 90% of the claims. But the nuisance suits are in the 1-5 million range, so we can get rid of those with sovereign immunity. Then you have the really bad cases and insurance covers those.
You are really going off the deep end to bring up the contention between the executive and legislative branches. Municipalities, quasi governmental agencies, etc. don't get wrapped up in that. They aren't somehow immune from the laws. The legislature can pass laws that the executive branch has to follow - it just has to be within the rules.
The GORs are rock fucking solid. Just because you don't anticipate something, you don't get to get out of a contract. If I buy a car, and 6 months later car prices drop 50%, I have no recourse because I didn't know the prices where going to drop. That's the risk you take entering a contract. Here it is worse ... they entered the GOR, which was signed by FSU for the express purpose to keep schools like FSU from leaving the ACC like Maryland did. It's doing exactly what FSU wanted it to do, they just don't like it anymore. Tough shit. Shouldn't have signed such a long-term agreement. Contracts often turn out not to be fair to everyone. If you don't like it, you can refuse to fulfiil the contract. The ACC would then get all the money FSU generated elsewhere. But FSU can certainly withdraw from the ACC, and they would lose about $500 million.
Promissory Estoppel doesn't apply here. There is an actual, fully negotiated contract with consideration.