Jan-March you won't find much to rent for less than a month. Beginning in April you can find a variety of homes to rent by the week for less than you would pay for a hotel or The Villages either. You do have security deposits but you get it back if you don't tear the place up. There are availabilities all over the place and here is one good place to look
The Villages 4 Rent, Rentals in The Villages. If it doesn't come thru: the villages 4 rent . com just remove the spaces in the url. You can search by village, price, amenities, whatever. I'd love to rent you our house but it is already rented through March and we'll be using it for a couple of weeks in April. It is property # 2060 in Tamarind Grove. Good luck finding something.