Apparently you don't understand how the Government program voucher works. The Government voucher programs (LINC, Section 8, FEPS, etc.) do NOT pay the entire rent that landlords want. The Government programs pay the entire amount that the program can afford. There is a huge difference.
Example: most NYC vouchers for a 2 bedroom is $1268./month. Landlords want $3,000./month for a 2 bedroom. $3,000. - $1268. = $1,732. Who is paying the $1,732. difference? Your government program voucher refuses to pay the difference and you don't have the cash to pay, so what happens? Landlords will reject you and take in somebody who can easily pay $3,000./month. Sorry, but there is no law that says landlords have to accept you. Landlords are legally allowed to discriminate based on finances.
What you need to do is get your name on waiting lists and try to get into NYC housing projects. Also look for available rentals that are up to $1200. - most likely a studio apartment. You may have more luck in the Bronx because apartments are way cheaper than Brooklyn.