I just searched AirBnB for Santo Domingo, in my budget of $550 for 3 months.. zilch.
I bumped my budget to $700 and still kind of depressing. I hope AirBnB rates are representative of the actual market there.
I'm not convinced that AirBnB is the right place to search (I own a hotel and know AirBnB very well). AirBnB tends to be focused on tourists for one or two week stays, so prices are comparable to hotels. Your new budget of US$ 700 per month works out at RD$ 32,000. I know in the Zona Colonial you can easily get a one-bed for that budget - the manager of my hotel has just moved out of a place with five bedrooms which she was renting for RD$ 33,000 a month. The issue is the three months I think. Dominicans tend to want at least two months deposit, and normally three. Then rent paid monthly. So if you are possibly thinking of staying longer than the three months, my advice would be to find a place on a one-year contract, negotiate to pay 2 months deposit (including the first month) then pay monthly. If you move out after three months, worst case you will have paid one extra month but will be paying rent at more like the local level rather than the tourist level. Just make sure that the contract says that you can move out at any time but forfeit the deposit if you do (most say that anyway).