I agree, Becky23. If it would be possible to go another day, I would avoid MK on EE day. However, if you don't have another choice you do have options:
*Schedule a Character Breakfast (my suggestion would be Crystal Palace) for as early as possible. They begin at 8:05, I believe. While this won't get you in before the resort guests, it will get you in a little earlier than the off-site guests.
*Go to Fantasyland first and grab a Fast Pass for the attraction you want to see, and then ride a fast loading ride while you wait...like Small World, which usually doesn't have a long wait.
*When your Fast Pass window is open, grab your second Fast Pass BEFORE you ride your first Fast Pass (the ticket will tell you when you can get another Fast Pass). That way, you can be riding a ride while you are waiting for your second Fast Pass to become available.
*To pass the time waiting for your Fast Pass, you can also take in some shows, as Steve has suggested.
*When things get really crowded in Fantasyland, move on to Adventureland or Frontierland.
*Or, when things get really crowded everywhere, cut your losses and move to another park (if you have park hoppers).
*Parade time will probably have shorter wait times also, but with your young children, that might be right during nap time!
Hope this helps!