The most economical way to transport the car is to go directly to the steamship line. Ocean carriers in Miami that have direct service both to Santo Domingo and Puerto Plata are Seaboard Marine, Anitllean Marine, Crowley, Tropical Shipping and Sea Land Service. Seaboard, Crowley and Antillean are the first choices. I have not checked the costs recently but your ocean freight should not be more then USD $500, but probably less. Check information (305-555-1212)with the telephone company or surf the web. All the carriers have websites and you could get their office information this way.
The big trick is getting the car out of DR Customs. You will probably want to go with a Dominican Customs House Broker to assist in this matter. They know the ins and outs of dealing with Customs (Aduana). Likewise, when speaking with the customer service departments of the steamship companies ask them for the information. They have it all and do not forget to ask for the name, etc of their DR offices.
The DR ports are corrupt so there is the possiblity of a bad experience. This is why I recommend you use a a Dominican Customs House Broker. If you go in by yourself you will be an outstanding target. Make sure your car is stripped cleaned of anything of value. As a comment, Santo Domingo has a way to many cars so make sure the government is allowing imports of your particular used vehicle at the time you plan to ship.
Good Luck and keep your guard up. General ZZ