As an attorney who handles these type of claims on behalf of insurance carriers, my recommendation is that you should NOT admit that you were in any way responsible. Gather up all your documentation and attach them to a professionally written letter requesting reimbursement for your complete damages. I would reference that you have not involved an attorney, but will be forced to so if you cannot resolve them claim to your satisfaction.
There is no doubt in my mind that the insurance company has already internally agreed to pay all your damages and the adjuster handling the file is simply trying to save his company a couple bucks