My car got into an accident and is at an auto shop. The claims adjuster came and deemed my vehicle as repairable. Then through more investigation, a supplement was requested and now my vehicle is deemed a total loss. I was contacted by my insurance representative to inform me that my car is a total loss and that I should release my car to be moved to their storage yard. Am I required to move my car even though they didn't offer me a fair market value for the car? If I do decide to keep my car at the auto shop until they offer me a price for my car, am I or my insurance responsible to pay the auto shop for any storage fee? What if they auto shop gave me a signed document stating that they will not charge me, personally, any storage fees? Can this be used legally if they were not to abide by their word, on the document?
You have the right to refuse the insurance company's offer and keep your car, but that would be self-defeating. You can keep the car at the shop, and you will incur storage fees; it is possible but quite unlikely that the insurance company will change its mind and offer you more money (unless you're contending that they made a serious technical error in computing the value). The insurance company has no obligation to the shop, but they might have an obligation to you (i.e. it might be a term of the policy that they will pay storage fees up to some limiting condition).
Whether or not you have to move the car or pay storage fees is between you and the shop. If I understand your idea about a signed document from the shop waiving fees, you're suggesting that they might waive the fees if you end up having to pay it, but they would stick the insurance company for the amount if the insurance company were to pay. Again, the insurance company only pays what you owe, and under this scenario, you wouldn't owe storage fees, so the insurance company wouldn't be obligated to pay. If you were on the up and up about this, the insurance company would just say "No, that doesn't work". If you were not on the up and up about the arrangement, that would be fraud.