Willie, I am not one of the legal beagles on this board, but I've been through this process a few times from the employer's end, as have clients of mine who are also friends, so I'll offer a few thoughts.
First, it is not clear to me whether the university has an insurance carrier that is involved, or if the university is self-insured via some sort of state group. I can't feature that WSU has its own self-insurance program, though I suppose it is possible; more likely if the state chooses to self insure the universities for this sort of thing it is through a central group in Olympia. The "outside carrier vs. self insurance" question is relevant in that if an outside carrier is involved, they will have a heavy say in whether the case is settled or litigated. If they can settle for what they consider to be peanut money, they will do so. Even if the state self-insures, there will be people in Olympia who oversee the program, and their role will be to behave in a much similar manner to the people who would be working at an insurance carrier. There is a third possibility...the state may self-insure up to a limited amount, and beyond that they go to a carrier who insures them only against a catastrophic loss. In that case, the goal of the attorney will be to settle near the high end of what ever the state will cover themselves, because unless the lawyer has a slam dunk case, the catastrophic policies will often go to the litigation mat due to the amount of money involved, and the risk/reward of a contingency pay day diminishes for the lawyer. The lawyer may have to guess that amount, because I don't believe that sort of information falls under the public information laws (for obvious reasons).
Second, as to compulsion. My experience has been that in most cases a policy holder cannot be bound by the insurance carrier to settle. However, every policy has coverage limits, as well as limitations on the carrier's liability in the event of misconduct by the insured. If the policy holder opts not to settle, and the award exceeds the coverage limits, they will be responsible for the excess. And...pouring salt in the wound...if the lawsuit's conclusion includes any sort of determination of misconduct by the insured (which is not only possible, it is actually likely in the event of a large award), then the insurance carrier may be completely off the hook and the entire judgement will have to be paid by the insured. (note: if WSU is completely self-insured, either by themselves or through the state, this doesn't apply...and it is the single biggest factor IMHO in whether the final result in the settlement decision would be the same for insured vs. self insured if WSU has any reason to think they may lose a case). For these reasons, the insurance company has a number of potential levers they can pull in "persuading" their client to settle...and that is often why settlements occur. Often the insurance company and insured will have a very strong case, but choose not to gamble on getting an unpredictable jury. The ambulance chasers recognize this, of course, and use it to their advantage. Further, they make massive contributions both individually and corporately to friendly legislators (who usually themselves are attorneys) to perpetuate and make worse the entire tort system. Long story short, it becomes another parasitical cost of doing business in that particular state.