Is your stroller a side-by-side? If it is, I wouldn't bring it. You need a skinny one to get through places. Hopefully you have one that is a double back and front. You really don't want to go through a big airport without the kids being in a stroller.
Now, if you have two umbrella strollers, and two adults, that would be good too. Light weight is good, and you can hang your stuff on the strollers as well.
Don't ship a box of diapers. Just pick them up down there in a store (if you are renting a car, that's really easy.) I would think that you could stash diapers here and there for the plane trip - stroller, suitcase, pocketbook, jacket pockets, etc. if you are staying on the grounds and not renting a car.
Take sippy cups and after you get through security, fill them with something to drink. Have them sip on them while the plane ascends so their ears won't hurt so much. Order children's meals for them in advance with the airlines. Bring snacks too.
The water at Disney tastes awful. Get bottled water before you get into the park. I hope you'll have a little frig in your room at least. If you can freeze the water, the bottles can be ice packs for a soft bag to keep your lunch in. Peanut butter and jelly sandwiches are great for lunch, along with fruit. If you can stop at a grocery and get that, along with cereal for breakfast and milk, you'll have breakfast and lunch taken care of and then you can go out for dinner. That's a great savings, compared to the expensive meal tickets. (For more opinions on the meal plan, check the thread about Disney written earlier today.)
Have a great time!
Dawn