Have done the airtrain with luggage from Newark deal. Given the ordeal I had just getting to Hoboken (not JFK), I’d be nervous about six hours.
Of course the best bet is finding out if American is transferring any aircraft from Newark to JFK, and asking the pilots if they’d mind giving you a lift. Too bad they’d say no.
Consider renting a car for the day. Go to Priceline.com for the car rental, you’ll get the best non-coupon pricing. Then, it’s all under your control and you have the option of grabbing lunch outside the airport.
I just looked at weekday rates. Lowest rate is $60 ($75 total) for an economy from Dollar - but they have mini-vans for $69 ($85 total) which sounds perfect for you.