Well, I highly recommend planning your itinerary - start with which days you'll be at which parks, and then plan some meals. Your kids are really young, so planning lots of sit down meals may not work best for you and that is ok. But, if you know which day you'll be at MK for example, maybe you'd want to make an early ADR for Crystal Palace to start off the day with the fun character breakfast or at maybe lunch. If you plan for lunch, then you could eat before you leave the park and head back to your room for your afternoon break.
Another fun breakfast is at Tusker House in AK. I think your kids would really enjoy meeting Donald, Goofy, Daisy and Mickey. Haven't been to the Hollywood and Vine buffet at HS, but that would probably be a hit with your kids, too. Or, Chef Mickey's at CR.
In general, with young kids, I really like buffets because there is no waiting -- you arrive, get your table then hit the buffet.
And, finally, to answer your original question, I think you'll totally be fine with $1,000 to $1,200 for your six day trip. Your kids are so young. My girls ate like birds at those ages...portions are pretty large, one child could probably share with you or DH or even split a child's meal with their sibling. I really don't think you need to stress or worry about anything in the food dept, there are lots of great choices.
Please be gentle with yourself. Parenting three small children is very difficult on its own. Taking them to DW is magical but also a lot of work. Have a solid plan and above all, the number one thing you can do to ensure great days for your family is to arrive well before the parks open and be there in plenty of time for rope drop. You can accomplish so much in those first couple of hours. Collect FPs as you go, take an afternoon break, and return at night and use the FPs you collected during the day. You will not regret being there at opening. It makes all the difference in the world!
