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For your budget, especially since it includes ALL utilities, I don't think you will find anything in a complex in Nassau for $1600, let alone something cheap enough to allow you to squeeze your LIPA bill into the budget too.
You may find something in a 2-family home, but that is still low, especially if you want to include the utilities in the budget. I know two families that live in 2 BR apartments in legal two-family homes. They both pay all utilities on their own, INCLUDING heat (their landlords installed separate heat and hot water systems for each unit). Neither are in fancy neighborhoods. One's rent is $1750 (Long Beach) and the other pays $1695 (East Rockaway).