My best tip for saving on food is never pay for breakfast at disney (except maybe once or twice). I see you're staying off site, cutting out one meal in the parks saves a lot in itself.
Scope out the quick service meals online, some are big enough to share. At quick service locations order off the kids menu especially depending on your appetite, the kids meals include a drink so if you or your kids want a soda thats an easy way to get out of paying the $3 for a cup.
Water is free at every quick service location.
When it comes to snacks go into the bakery shops and DON'T go to the counter with the fresh treats. Disney has proportioned snacks that are exactly the same and you can get enough for your whole family for a little above the cost of the ones in the chill boxes. My favorite are the rice krispy treats. For $3-5 you can get the fancy character decorated ones, or for $8 you can get a bag of 8 rice treat balls. They won't have icing or cute shapes but they're DELICIOUS and also fresh.
Look for a list of free offerings, you can walk up to any of the flavored pop corn stands and ask for a free sample before you commit to buying a bucket full it's also a great way for a free quick snack.
The best tip though, don't save too much. Saving money on your trip can be easy, but you don't want to cut corners on your vacation.
Definitely try and have a picnic lunch in the hub grass though

It's beautiful and you'll make a lot of good memories there.
If you change your mind about table service try a lunch, they are worth it. Lunch menu's are usually similar to dinner menu's but they tend to run a little cheaper at some locations.