We've been delivering to the US (from the UK) for a good few years now, and have seen the level of credit card chargebacks inexorably rise - customers are swearing blind that they haven't received the goods.
Does anyone know of a carrier that requires a signiture from the customer before releasing the delivery, or otherwise offers proof of delivery?
Whatever happens don't get personally involved in this lest you have a wish to descend into a dark bureaucratic pit of despair from which you may never recover. For the sake of your sanity I urge you, get someone else to deal with carriers.:)