I have looked into the California case law and other authorities, and I cannot find anything directly on this point. The general authorities I have found indicate that you simply have to offer the same type of proof that you would offer in court, which could be your testimony if you have nothing else.
You probably want to make sure that you have a copy of the policy (the entire policy, not just the "declarations" page e.g.) and see what it says about how you must prove your claim. If the adjuster insists on having a receipt, then you may want to ask him or her to explain to you why they are requesting this, especially for such old items. They do have a duty to deal with you in good faith. It is usually best to not start with threats about your recovering punitive damages from them, but if you have a valid claim, then ultimately they should be concerned about such a claim.
If you were to pursue the claim in court, your sworn testimony would be sufficient to assert your claim. Of course, the finder-of-fact (e.g. jury) could accept or reject your testimony.
If the policy allows the insurer the right to take your "examination under oath," then I generally think it's a good idea to offer to allow them to take such an examination if they have questions about the loss; it tends to make the insurer look more unreasonable if they refuse your offer.
I am sorry that I don't have anything more specific to offer for you to use with the adjuster. Please let me know if you have any more questions.