When my kids were little, I used to buy Disney shirts at the
Disney store when they had them on sale (like $4). I would get them a new one for each day at the park. When they wore their brand new Disney shirt each day, they didn't seem to want much in the way of souvenirs??

They felt they got something each day. I never called it a souvenir, but since they got something new, they were happy. I did let them pick one toy or something each trip. Something like a Disney princess doll or a mr. potato head. But if you give the kids a budget up front - like $30, it helps with the "gimmes". I usually made them wait a few days before buying anything. Also, Walgreens has some Disney light up wands that you can bring with you. My kids never were tempted to get the light spinners at Disney because they had their own light up wand. I also brought the light up bracelets/necklaces that I got from Michaels for $1 per pack. I still bring these and my kids are 19 and 15! We often share with little kids around who are getting restless waiting for the show.
Other people have given great ideas about food. You can order kids meals at counter service places. Although I'm not sure how great that is anymore since they changed most of the kids meals to the same boring, yuck. But usually the hotel counter service places have better kids meals.
When we go to table service places, sometimes me or the kids order an appetizer as our meal. Like DD might get the potstickers at Yak and Yeti as her meal. I like to get the Pita Quesadilla appetizer at Rainforest Cafe as my meal. At TRex, I got the add on ribs as my meal (added on to DH's meal) and then shared fries off my DDs plate since she doesn't really eat fries.
I also bring snacks like nutri-grain bars or packs of nuts to have as snack if we got hungry between meals. We still got a mickey bar or dole whip once in awhile, but having the packaged snacks helped.
It might not save money, but I found that getting those little activity books from walgreens (3/$1 sometimes with sale coupon) and having them in the backpack helped keep them occupied while waiting for Fantasmic or the parade, etc. They have Disney themed ones available. I would bring a pack of crayons too.
I see you might be staying at Swan/Dolphin. You might want to check into the Restaurant.com website because there are several coupons there for the restaurants at Swan/Dolphin. Don't pay full price for them. They have coupon codes each month where you can get these for $2 each. Also, the Swan/Dolphin has a character meal, I think. Its smaller than the other Disney ones (I think only 2 characters), but it is cheaper and might satisfy the kids enough.
Maggie