DHCR administers or provides access to many different programs. If you meet a program's eligibility requirements, you can pursue just about any house within your approved price range and comfort zone. For example, the DHCR website contains excellent information on five separate SONYMA mortgage programs. Most houses will qualify for every one of these.
Your best bet, is to retain a skilled buyer's agent who will, first, help you determine your wants and needs in a house; and then second, move heaven and earth for you to find ALL of the resources that will help you secure the wonderful benefits and satisfactions of home ownership!
If you'd like a free, no obligation consultation about practical plans for homebuying, just Email me at RCoake@Rochester.rr.com. Or phone me at 585.223.4449. Best wishes!