Hi Ashley,
First off, welcome to the Forum and Puerto Rico soon. I am a little confused about your statement, you want a long term rental but are thinking of moving around. In my book a long term rental is 6 months to a year, not a couple of months.
Airbnb is great for a few days and maybe a month, it is very expensive past that point. I disagree on looking for Vacation Rental units, they tend to be thousands of dollars per month just like Airbnb.
A realtor can help, clasificados can also help and the people in the Facegroup "Friends of Rincon can help even more. They are feet on the ground, they drive around and see units with signs for rent (Se Alquila).
Once you are in the area, drive around, in Rincon you should be able to find a place to rent in the 600-900 dollars a month if you deal directly with the owner. Stay in an Airbnb for a week or two while you find that inexpensive rental unit.
Take a look at this: