I have to agree with above poster. $1500 should just about do it. I usually tell people budget about $40 to $60 per person per day. I also tell them to buy/bring breakfast foods (cereal, pop tarts, muffins, milk, juice) and their own snacks (water, pot. chips) as this does cut down what you will buy in the parks. Most people will usually eat at least one meal as counter service but that gets old when your going to be there for 10 days at a time. So I would say add in a few buffets along the way to break up monotony. Instead of having dinner as a buffet, you may want to do a late lunch buffet (same food, smaller price) say around 2:30 in the afternoon. If you time everything right from the morning and eat a nice breakfast, snack in early afternoon and then big lunch around 2:30 you can find something small to eat at night or even split a meal if everyone isn't hungry. This also helps to keep the budget in check.
Additionally as the above poster stated, I wouldn't set aside too much for souvenirs ($250/$300 tops). A great way to save on this is not to purchase anything right when you see it (impulse buying is the worst). Write down what it is, the amt $$, and where to find it. During the last 2 or 3 days of your vacation, sit your family down, review the list to determine what you really want off of it then go get those items and your done. I find doing things this way allows me to really choose what I would like as a souvenir and I usually knock a few things off the list after really thinking about what I would do with it once I got home.
Have a great trip.