I'm budgeting about $200 per day for our family of five, and one of our kids is 13. So I pay adult prices at buffets for him, although he eats off of the kids menu at most sit down restaurants.
We generally do breakast in our room, one sit down meal and one counter service. At sit down restaurants, we might share one appetizer, and we usually don't do dessert, although we do get non alcoholic drinks. We're not big snackers.....we usually share something.
I would think $200 would be fine for a family of four. How old are your kids? If your kids are 9 and under (paying kids meal prices), $350 would be way more than you need.
For example, Chef Mickeys would run you around $98 with tax and tip, But other buffets would be less. Vouchers at AK would be $36, and that would get you lunch and a snack! You'd spend about $30 at Cosmic Rays (counter service) or maybe $70 at Cindy's for lunch.
Julia