Anastacio, I don't know your background or familiarity with Santo Domingo. Do you speak or read Spanish? Are you looking for an apartment , or just a room, or something else? Are your temporary accommodations just for you, or for a family?
Maybe you don't want to run around the city looking for signs in areas you like, or maybe the areas you like are priced out of your range.
Looking at today's Listin Diario, there are several ads for rooms for rent, some are two rooms, all between RD$4000 and RD$20,000 a month (US$110 to US$550 a month). Many times, the owners actually prefer foreigners as renters because they know they'll leave without a fight after the month or two is up. If you can't read Spanish, that does you no good without help. If you don't know Santo Domingo well, you might be looking at renting a room in a real garden spot with real problems.
Most apartment or house renters in the DR are forced to commit for a minimum 12 month term, and owners want 3 or 4 months rent in advance. And they get it.
Are you are looking to rent right now (August) just when many Europeans are starting their holidays? Where do you think they go? Many come here, and do exactly what you want to do, which is find a cheap place to stay for a month or 6 weeks in Santo Domingo, close to things, etc. where they can spend their vacations drinking coffee in the mornings and beer all afternoon, and looking at (or chasing) pretty girls passing by.
Go for a walk down El Conde in the Zona Colonial and find some foreign looking older guys sitting out in front of a cafe nursing a cup of coffee or a beer. Ask them if they know where you might find a good deal on a place to stay for a month or two, with the characteristics you are looking for. If you don't need a place until September, maybe you can grab where they're now staying.
If they're from Sicily, they might not speak English, or ignore you, but there's a good chance they're staying in a relatively cheap place with conveniences nearby.(no offense intended for Sicilians, but I have watched these guys for years and they are kind of clannish down there).
If they don't know, they might know someone else who does. -0- cost to you, unless they hit you up for a beer, or try to slide in a commission with the owner of the place they refer you too.