Andy,
Yes, it may cost more but I think that I would call a car service and have them meet you either outside security at JFK, or on the street(they'll charge more to meet you inside but will have your name on a sign). They usually charge a flat rate from airport to airport, their cars often are a bit newer and cleaner, the air conditioning will work and the drivers will likely not be listening to music you hate or talking on their cellphones.
Just to give you an idea I went to Carmel Limo website (
http://www.carmellimo.com) and typed in a request for JFK/EWR and here are the booking choices that came up (if you want inside pickup it's US25 additional). If you want a newer vehicle choose Luxury Sedan.
Have a good trip. Regards, Chuck-
FULL SIZE SEDAN (1-4 Pass) $70.0
--------------------------------------------------------------------------------
LUXURY SEDAN (1-4 Pass) $77.0
--------------------------------------------------------------------------------
DELUXE SEDAN (1-4 Pass) $77.0
--------------------------------------------------------------------------------
MINI VAN (6 Pass) $85.0
--------------------------------------------------------------------------------
STRETCH LIMO (8 Pass) $130.0
--------------------------------------------------------------------------------
STRETCH LIMO (12 Pass) $180.0
--------------------------------------------------------------------------------
LARGE VAN (14 Pass) $115.0
--------------------------------------------------------------------------------
SUV STRETCH LIMO (1-14 Pass) $185.0