Food Courts and Counter Services should certainly be within your $10-12 breakfast and lunch meals.
But....if you're looking at Table Service meals for dinner, then $20 may be on the low end of things for many of the meals. As Pumbaa pointed out, you should definitely go check out the menus. There are a few character meals that are $18.99 for breakfast......but then of course your $10 breakfast budget would be stretched very thin to breaking to translate over to a dinner counter service. Most of the character meals are mid-to high $20 for lunch/dinner...and the ala cate menu meals are going to go over $20 except for the very lowest cost item on the list (remember that ala carte menus at Table service don't include a drink or dessert or appetizer like a character buffet would). I'm not saying everyone has to eat dessert and/or appetizer, but at most of the TS a lowest cost entree and drink are just going to reach your $20 (and don't forget tax and tip for TS meals).
Now....my family also consists of 1 "real" adult and two "Disney" adults.....and my daughters and I very often share two meals between the 3 of us (at CS and TS). With that method we could come pretty close to your food budget. But...my kids are girls and light eaters.....you have boys and friends tell me their 10 to 12 year old boys are big eaters so if yours are too, sharing might not work. We also typically stay offsite so we have a refrigerator to have breakfast (yogurt, toast, cheese, etc) for the days we don't do a character breakfast. We also bring in our own snacks so we don't have to spend $4 or more for a snack, each.
To give you an idea.....Trails End at Ft Wilderness is considered the "bargain" of all of Disney because it's the cheapest buffet on property. Breakfast buffet is $12; lunch is $13 and dinner is $19 (before tax and tip). The food is decent, homestyle comfort food not fancy stuff, but fills you up for little money. As you can see those prices are right at your budget, so being that it's considered the cheap joint, most of the rest will be a little over budget for you. But....Ft Wilderness is an easy boatride from Magic Kingdom, so spread a breakfast here, a lunch there and a dinner over the length of your trip and you'll help save a few bucks for the nicer TS meals you're wanting to do.
What we always do is while planning which parks we want to go to, we'll also make our ADRs for TS meals....and have a good idea of what CS we probably want to eat at.....then hit the menus and pretend we're going to order right now to get an idea of what we need to budget. It's not carved in stone and sometimes we hit a different CS.....or order something different, but for us we're usually pretty close to what we actually do.....and if you are torn between two entrees, I always budget the higher one of course.
The other thing that we have found is that even though you'd think with all the walking/exercise that you'd be famished at meal time.....for some reason we aren't, maybe it's that our stomach is distracted by all the fun.....but we typically don't eat 3 meals a day at Disney. I tend to schedule our breakfast buffets for "brunch" more than early breakfast...and that takes care of both breakfast and lunch.....if we wake up a little hungry we'll grab a quick yogurt to tide us over.....so we only need two meals a day. Not ideal health wise, but then you're not exactly eating healthy either, lol. Trail mix, heavy on the protein packed/hunger satisfying nuts and low on the sugar spike/hungry right after type snacks also help keep us going.
