Newly made family of 6 here. Here's what we do:
1. Southwest Credit Card: if you're flying (and have good credit), you can apply for both their plus card and premier card at about the same time when they are having their 50,000 point promotion. That'll give you 100k points for the cost of the two annual fees (~$180) and the security taxes and that should be very close to enough for 6 tickets if you book when the window opens up.
2. Purchase tickets from undercover tourist: if you can do it ahead of time, even better. Not sure what your military discount is, but that might be a better option.
3. Unless you're really stuck on staying on-site, I'd recommend an offsite condo or townhome at Windsor Hills or a similar resort. Everyone gets their own room (or at least the parents do) and we've stayed there for a little as $600/wk. Even when you add the cost of parking and a car rental, it's still a better deal.
4. Staying offsite has its advantages in that you can easily do breakfast and lunch at the condo, allowing you to save a ton of money on food.
Currently, not counting food, our budget stands at $3,500 and that includes airfare for 6, 7-day park hoppers, 7 nights at a 3 bedroom townhome, and a one week stroller rental. Biggest chunk of the budget are park tickets.