I am sure you could get a fairly decent furnished apartment in Pueto Plata for 10000 pesos per month if you are prepared to live among Dominicans. This will almost certainly subject you to noise issues which may or may not be a problem for you. It may also give you security problems. That is harder to assess without knowing your life-style and the exact location. You will also need to be patient with recurring problems with water supply, plumbing, electricity, car-parking, rubbish disposal, and so on, though these are not necessary life-threatening or even happiness-threatening if you can attune your expectations to the Dominican way of life. If you have a short fuse, don't try it!
You might consider renting unfurnished and gradually import cheap furniture. Would give you a much wider choice of apartments as Dominicans don't rent out furnished much.
As for Sos?a I would think you would be lucky to find anything in El Bat?y. You might need to consider Charamicos, Abajo, etc.
If you want to live like a gringo among gringos you might need to pay more.
