If you are staying onsite, then as soon as you get to the resort, ask where you go to get a printout of your ADR's. I have done this for each of my stays at all the resorts. Then as others have said, show up for your ADR, if they make a stink about your party number, whip out that sheet of paper and show them how some of your ADR's are for the right number, some aren't. Since you are a family with kids, obviously you aren't planning on dining without some of them or adding two more later in the trip! That printout will show the restaurant that the mistake was Disney Dining's and not yours. They should then seat you. And if the hostess says he/she won't, ask for a manager. Don't leave that podium, don't get angry, just stand your ground. Disney aims to please. Its easy for some person at Disney Dining who will never see you face to face to tell you sorry, no can do, we can't help you. Its a different story for the cm at the restaurant who has to deal with the problem right there. That cm will work harder to make you happy believe me!
For what its worth, this same thing happened to me. I had a party of 7 and when I got my printout it had us down as a party of six a time or two. I just showed up, told them I was there, they said "party of six", I said, "Nooo, party of seven" . They shrugged and sat us right down no problem. I think this happens quite often.
Don't worry and don't let them turn you away! And I agree, no more calls to Disney Dining, they dropped the ball, now the restaurants themselves will fix it.
I also just want to add, that the most important thing is you have a reservation with a confirmation number. If some of your reservations just didn't show up at all, that would be a much bigger problem. Because then they would have to find one for you and it sounds like they are all full. But since you have a reservatioin, then the restaurant will shuffle to make it work with the number you have. The one reservation that has you down for a larger party in particular is not a problem. The one that has you down for four when you are five is also probably not going to be a big deal either. I really think you will be fine! Just get the printout and have it with you to show the restaurant the overall problem. Again, they are used to Disney Dining making mistakes and then having to sort it out!
