In Puerto plata, try Esquina de Cachita; 586-8372 In Santo Domingo, try El Conuco; 686-0129
Eat lots of Sancocho, Lamb?, Mang?, Bacalao.
I like to eat food from the street vendors, but some people are afraid to, because maybe they will get sick. I go to the DR about every 2 months, and I eat everything I come across and never get sick. I don't drink tap water though.
If you are daring, go to Boca Chica and eat some of the seafood they always sell on the beach fresh. It is very tasty. Agree on the price before you accept it, as some vendors will try to get over on you.
Since you are coming from far away, you might have some initial mild upset, but just go to a "farmacia" and tell them what ails you, and they will fix you right up.