If they misspelled your name, or got your birthdate wrong, it's easier to just live with it. Correcting one of their errors is the same as entering hell. For most people, it's not worth the long, hard fight. (or expense)
Also, many dominicans have the wrong birthdate on their birth certificates because papa didn't register the birth within the first 30 days, and without doing that, a dominican cannot get their legal documents needed later in life , SO papa just tells them a different date within the 30 day time limit when he finally DOES go to register his newest child's bithdate. Barring that possibility, it also happens if they later show potential at baseball, jaja.
If you changed your name legally in the US, you must present the court order that did just that, getting it translated and notarized, and probably certified by a Dominican Consulate inthe US before presenting it here for the changes.