I'm no expert - simply a casual market observer, but from what I can tell, your budget is about 30% below market rate for a patio villa with those extras during that time period. You may be able to achieve that rate were you to rent for a longer term, perhaps annually.
I hope you find that for which you seek.
EDIT: I found this one for $2000/month for those months, but you might opt to rent a golf cart from The Villages Golf Cars as it does not appear included.
www.rentfromavillager.com/3201
EDIT - EDIT: I also found this one at $4993/month and includes a golf cart in the Village of Newell:
www.rentfromavillager.com/2055