so, I don't receive food stamps, but I got myself into a pretty bad financial situation a few years ago, plus my husband paying a huge chunk of money for child support (which, I totally agree with, don't get me wrong!!), we have very little money to spend on groceries.
we buy a lot of things in bulk or on sale like pasta, rice, couscous or a couple pounds of chicken breasts at a time to be frozen or fish (a kilo of salmon or tilapia is way cheaper than just buying a few fillets) or a huge lot of eggs. Some veggies and fruits that are more expensive than when in season we buy frozen. The things we only buy fresh are veggies and fruits when necessary and we change what we need based on season.
I have no idea how much you receive or what you buy as "healthy" foods. Pre-prepared and packaged is always going to cost more so we prepare almost everything ourselves. You are right with smaller portions and more exercise but obviously you want to get your fruits and veggies in too.
Do you cook? Making soups yourself and freezing portions for when you want them are much cheaper and healthier than pre-made soups (assuming you don't dump as much sodium in or use preservatives), finding ways to use up leftovers (I use leftover pasta in salads with tuna or eggs or even cottage cheese) means you can make things in bulk and then have them on hand to make lunches or even dinners. Making your own pasta sauce is easy and can also be done in bulk (and tinned tomatoes are more than perfect to use and can be cheaper than buying fresh). That way you could take a Sunday for example and then just cook and pre-make and portion things out for when you need them during the week.
Like I said, I don't know what your amount is, but we really also have limited funds for food. Sometimes I feel sorry for myself and wish I could have a treat or something extra, but honestly? Living this way has also helped me to lose the weight this time around. There have been no take-outs or calls for pizza - making my own chinese is healthier and I just whipped up some pizza bread the other day that was diet and wallet friendly.
I hope you find a solution or that I could help you just a little bit