I may be wrong, but I think that because the Dolphin (and Swan) are not technically Disney hotels that they do not transfer luggage for you. You will have to call a cab to get yourself and your luggage over to the Dolphin. I don't think that a cab will cost that much since the resorts are not that far apart.
I suggest calling the CS (do you mean the Coronado Springs?) and asking the front desk people if they have a way to transfer luggage or they could suggest a cab company. I know that Disney does transfer luggage between their own resorts and people can get around to all the resorts using the buses/monorail/boats, but it does take some time with the roundabout ways you must use to get from one resort to another. I just am not sure if the Westin hotels participate in the luggage exchange.