Welcome Mike to CMC. You will find lots of information here.
My husband & I retired in December and moved here to Cozumel permanently in January with our 2 small dogs. We have met folks who come for the winter and bring their dogs. Not sure about doing that for just a month.
We are renting right now while we look for a place to buy. There are lots of property managers on the island and I think if you do a Google search for "long term rental" you will find a few places.
I know folks here who come every winter and they line up their rental again for next winter before they leave...but not sure if it is too early for brand new rentals. I will let the helpful folks here post an answer to that.
Where are you from?