When I have purchased travel insurance with USAir/American, it has been with Allianz. I do not always buy it - but sometimes, particularly in years past when I feared that something could happen with family members or a legal matter requiring me to cancel the travel. I have made two claims over the years for reimbursement because of unrecoverable expenses incurred before a change of plans and Allianz paid both very quickly without any hassle - but both times I submitted documentation, including bills and receipts.
I would guess that your insurance is, as Carol said, not with Delta directly but with a third party such as Allianz. In any event, what you need to do is the same.
Find and call the phone number you have for the insurance and persist until you talk to an actual human who is handling your claim - this may take some time. Find out what you can do to support the claim. If nothing else available, submit affidavits of anyone and everyone (including the two of you) who can attest to facts supporting claim, along with bills and receipts for any money spent.
If claim still denied, you can contact your state's insurance department (just google your state's name followed by "insurance department" and it should come up with a website telling you how to file a complaint) and file a complaint. See what happens with that.
Last last resort is to consider legal action - if you think it cost and time effective. Be careful of losing more money than what you already have lost.
Good luck.
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