Good Morning Al,
The Closest park that could handle your rig is
Blydenburgh County Park - Smithtown, I've seen large rigs in the park many times. Blydenburgh is kind of my "home" park, it's about 20 to 30 minutes from work/home, so I can camp there even on school days.
Smithtown and Huntington share a border as you can see on this
map, run your mouse over it and the town names appear.
Bear in mind that the county parks offer no hook-ups. The do allow generators during non quiet hours. Some sites do have water nearby, in "hoseable" distance, with a Y connector you could most likely stay connected to water at least. You do need the Tourist Green Card as outlined above.
As far as the "best" way in, that varies with construction schedules and time of the year - your route for example is better if the Yankees aren't playing. Driving onto or off of Long Island is always a burden during normal travel hours, things get better later at night (but now we are starting to do overnight constuction - so that screws things up a bit) or very early morning.
I'll have to talk to some buddies about how they get their rigs on and off the Island and get back to you.
Mike
Addendum: Blydeburgh does have a Dump Station Fee Charged for use.
Noticed that the "mouseover" feature on the map of Suffolk County I mentioned above doesn't function in all browsers so here is another, that includes town names
Suffolk Town Map