Linehand,
Air down? You need fatter tires! Just kidding. I grew up in Montauk and have driven thousands of miles in the sand. Generally the lower the pressure the better. Keep in mind that blowing a bead in sand sucks! I usually go down to about 20 or so. In the sand momentum is everything. Gradually build up speed; gradually ease off the gas. You NEVER need to touch the brake pedal. NEVER!
I had a 1955 Dodge PowerWagon with **** Cepek 38.5 x 16.5 tires...15" wide that floated on sand at 30# in the rear and 35 on the front. I drove over sand covered outlet from a pond and promptly sank to the axles.
It's not a question of if you will get stuck; rather when will it happen...
On a side note I hope you did not drive in the dunes. HUGE fines, and the ecosystem is fragile.
You also need town stickers for most beach driving, but also state stickers for some beaches as well as county stickers for a few. The East Hampton Town Police love to give out $500.00 tickets to those not properly permited...
Contact the LIBBA...Long Island Beach Buggy Association for info.
Now as for driving on the multi million dollar lawns...Woohoooo...
Eddie