Claiming on Disney's Liability Insurance

Has anyone ever had to claim on Disney’s liability insurance? Some of you may remember from Chat that I fell and broke a finger at Disney in April while chaperoning a school trip. Although their follow-up call gave the distinct impression that they had no coverage for liability from guest accidents, I’ve since been informed that they are self-insured for the out of pocket amount I have to pay outside my health insurance.
I am not seeking any punitive damages, pain & suffering, etc… I just would appreciate their stepping up to the currently $4,000 of out-of-pocket expenses. The school’s insurance gave me the contact information for Disney’s adjuster. I have now left three messages in ten days without any reply.
Does anyone have any helpful tips or experience? I’m not really interested in getting an attorney involved because I doubt they’d be interested if I’m not looking for additional money. (Also, having worked in P&C insurance, I tend to feel like people/attorneys seek far too much once they get involved.)

What caused you to fall?

I just tripped, so there’s no punitive liability or negligence involved.