When it comes to the groceries nowadays, what most people pay for is convenience. They're willing to spend a bunch of extra money just to save a couple of steps.
Most of the time (not always, but usually) its cheaper to avoid these things. Take chicken for example. If you buy boneless, skinless chicken breasts, they're $5/lb. But if you buy bone-in, skin-on chicken breasts, they're $1.50/lb. And they are super easy to do yourself. I tend to buy thighs more than breasts, fix a bunch up all at once, and grill them all at once. Then I can eat them throughout the week.
And the previous advice to buy the frozen vegetables is good, but you don't need to pay to buy the "Steam in the bag" type. Just poke a few holes in a bag of regular frozen vegetables and microwave that, works just as well, and is much cheaper.
Go for the store-brands. They are just as good (sometimes better in my opinion) as the name brand, and usually tons cheaper. But take your time at the store, look at the price per unit or price per ounce. Find the best deal.
And when it comes to fruits and vegetables, don't think you have to buy fresh to get healthy. Usually the frozen are flash frozen at the peak of freshness, which keeps most of the nutrients locked away. The "fresh" vegetables could be sitting around for weeks, losing the valuable nutrients.
Canned fruits and vegetables will have slightly more lost nutrition, since the processing heat will destroy some. But a good way to get the most out of canned foods is to not strain off the liquids, since they also contain lots of nutrients.
I like to just serve the canned peach, pear, or fruit cocktail juices over ice and drink that. Great in the summer.

Also, look to the less used fresh vegetables, such as collard or mustard or turnip greens. I love them personally, and they are absolutely packed with nutrition. Beets, when done the right way, are delicious too, although they are slightly more expensive.

And try to look around, look in your local newspaper ads for special deals and sales. I shot myself in the foot last week when I bought about 3 bottles of ketchup to "stock up" for $2.50, which I thought was a good deal since everything else was about $3, when the same store ran a special on the same brand for $1 a bottle a couple of days later.

I really do hope this helps some. Although I don't have a family, I also don't have a bunch of money. But I can tell you for sure, you need more than turkey sandwiches for breakfast, lunch, and dinner to have a balanced diet. I do like the idea of the rationing of snacks, thats what my parents did with me and it worked well.
And do stick with water. You're already paying a utilities bill, so use it. Much cheaper and healthier than other drinks. If you do need something sweeter to drink, go for the frozen concentrates. They're tons cheaper, and still taste great.

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