Am I responsible for storage fees on my vehicle after an opening offer has been made

The answer depends almost entirely on the terms of your insurance policy.  Your policy is basically a contract between you and your insurance carrier, which determines who is responsible for what in the event of a claim.  So, the first thing you should do is to pin the insurance company down and ask for a specific reference to a clause in the insurance policy that makes it clear that you are responsible for the storage fees in question.  In the absence of a clause that makes it unambiguously clear that the storage fees are your responsibility in this scenario, your next step is to remind the insurance company that courts consistently interpret ambiguous or uncertain language in insurance policies against the insurance company and in favor of the policy holder.  Next remind the insurance company that they have an obligation to act in good faith as they go through the insurance adjustment process.  Most likely the insurance company will back down.  If they don't, ask to speak to someone higher up in the food chain.  There's a bit of brinksmanship in all of this.  The sad reality is that insurance companies these days are not in the business of helping people.  When there's a claim, their business is to pay as little as possible as late as possible.  Their first response is to deny claims and make absurd demands like the one you describe.  You just have to stand up to them.