So......Have I lost my mind to think I can do it on that Budget?
Maybe your mind isn't entirely lost, but with three kids -- none of them really small -- you're going to be cutting your budget to the bone.
1. Don't buy tickets for every day of your stay. First, you can't afford to do it on this budget, and second the kids'll probably need a break mid-week. If you buy 4-day park hopping tickets with 4 "extras" visits, you'll be able to have four good, solid days in the parks PLUS four visits to water parks or the other smaller venues. Use the "extras" for your slow-days; sleep in, then hit the water parks. This'll cost $1213 from TicketMania, an online ticket discounter.
That's essentially half your budget gone. Only $1287 remaining for gas, room, and food. How far are you driving? You can't do much about the price of gas -- what does that leave for food and room?
2. Consider that in off-season, you may have the benefits of short lines and lower crowds (in theory), but no one ever talks about the bad side of that deal: The parks close early. They roll up the carpets as early as 6:00 PM! Parades and fireworks don't happen every day either. I'll take summer anytime -- it's not hard to plan around the crowds.
3. On-site, off-site -- they both have their pros and cons. If you stay onsite, the biggest "plus" is Disney's transportation, which is free and runs like clockwork. It means that your family doesn't always have to go to the same place at the same time. The negatives, of course, are that you don't have as much space in your room, and you end up spending more on food. If you choose off-site, it ALWAYS takes longer to get to the parks than you expect; two miles away can be an awfully long drive in Disney traffic. And don't forget to include parking in your budget -- I think it's about $8/day now.
4. Food is an easy place to save, though with such a small budget, I'm not sure eating in the parks once a day is realistic -- unless you really luck up and find a rental for $300/week. The cheapest meals in the park will be kids' meals, which run about $5 and include a drink. Anyone can order these, but it'll be repetative: chicken nuggets, burgers. Plan to bring as much as possible from home.
At this point, these are your biggest decisions:
How many days of tickets to buy?
Where to stay?
Once those decisions are made, then you can fill in the rest.