I also live in Queens and think that I do a pretty good job shopping on a budget. I don’t have food stamps or assistance, I’m not sure what you are able to spend but ill try to give some tips!
Sign up for Aron’s and seasons daily circulars. Seasons has a reputation for being super expensive. The seasons in the town I grew up sells most of their briskets Etc for 16.99 a pound, and in queens you can find for 6.99. Aron’s send coupons almost everyday. Te other day they had chicken cutlets for $1.99 a pound. Often I download the entire circular so that I am prepared when I get to the store. Aron’s has an outer area that contains some fruits and veges I’m sure you know. I found that they are much cheaper than the ones inside! A few weeks ago I decided to compare. The peppers are over $1 cheaper outside as are the onions and bagged potatoes.
Seasons sometimes has half off deli prices. You pick why you want and they weigh it, then cut the price in half. This doesn’t happen so often but when it does, if you have the room it’s wise to freeze chicken, prepared meats Etc.
Do you like sushi? I do! Main glatt on Main Street has excellent sushi for around $3.95 a roll. Sometimes I will pick up 5 rolls for dinner or my husband and me and its delicious and not expensive.
Meal Mart on main also has great prices that are sometimes overlooked. Their fried chicken is amazing and really inexpensive and they have huge portions. Good luck!