Few people do month to month, most want a contract, typically 6 months to a year. Have you checked clasificados? I put the URL on the site several times.
Funny how you were looking at Dorado, Bayamón, Ocean Park and jumped all the way to Humacao.
You can also go with a realtor.
You would be best to come look and rent what you want, lots of places do not have pictures or only have a few picture so you don't have the whole story.
Another option is rent an AirBNB which is cheaper than hotel for a week or two, go look for a place to rent, get electric and water connected then come back to the states to pickup the family.
WARNING: Most houses and apartments rentals are basically the wall, floor, and a ceiling. No furniture, a lot of times it may not have a stove or refrigerator. So you need to get those or sleep on the floor for a few days and drink warm water.