No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Hmm not sure if this will help at all, but it's worth a shot (maybe). Since you're in NY, I don't think you have a BB&T there. But they have a loan program called CHIP. You can do a search on the boards for more details, but it's 100% financing no PMI. Minimum score does have to be 680 now, however. It's basically their alternative to FHA. The max loan limit is $417K. Income restrictions apply, but it's based on your county.
Hope this helps
Maybe I'm confused, but if you have $50K to work with, can't you just buy down the loan to the $271K limit?
UPDATE: After posting the below, I saw that you stated your city and state so I looked up your county (Monroe County) and yes, the limit is $271K for a single family.
Are you sure the FHA limit is still lower in your area? When the increased FHA limits expired on October 1, 2011, the loan limits went down in my county too. However, the government then restored the limits in November 2011 and the loan limits went back up to what they were earlier in 2011. I thought this was the case everywhere and the loan limits for all counties nationwide were restored to pre-October 1, 2011 level. Here is a link to the HUD website where you can look up your county: https://entp.hud.gov/idapp/html/hicostlook.cfm