Clinic Abreu though they might find gonorrhea not the one you are worried about. I am assuming you mean in the capital since you didn't say and it is a big country. There also should be clinics run by NGO's with STD tests, condoms, the pill etc., usually there are.
Any decent doctor can refer you as well, really a hospital is overdoing it.
Clinica Abreu is probably the best or one of the best clinics/hospitals in the capital. An NGO is a non-government organization, e.g. Planned Parenthood. I imagine there are STD and birth control clinics in the DR run or sponsored by NGO's to get tests and birth control to girls (primarily) who can't afford it. There certainly are in Nicaragua where I spend my time, giving these tests and contraceptives for nearly nothing and in some barrios for totally nothing.
Any decent e.g , non public hospital, can test for VD and HIV. So can a doctor. You don't have to be a "real specialist" to take some blood and pee and test for VD and HIV and it is unlikely a doctor would take the blood or do the tests since that is what lab techs are for.
What is all the fuss about anyway? And where are you? This sounds sort of ridiculous if you cannot find a hospital or doctor, unless you are broke. Then look for the NGO clinics.