Hi Amy,
I have just looked in to this recently. AA is the cheapest I have found and will carry a dog to DR from the US for $100 as long as it is accompanied (probably pay full fare for your own ticket).
However, be aware that no US / UK airlines carry dogs between May 15th and Sept 15th due to the weather.
Additionally my vet told me yesterday that for dogs going abroad from the UK, the time between the rabies shot and the dog being allowed to travel is 6 months. Not sure if this is the same for the US.
Good luck.