We are a family of 6 and staying on site is just NOT in our budget, regardless of the discounts!
However, I think staying off site is a great idea and many resorts have a cool pool/kids play area/etc.. There are even houses with private pools that are available. I think that renting VRBO might be the most economical way for your family. Lots of nice resorts and private homes are available that are very close to the parks!