Hi dlwalt,
I do not know how much time it took you to gather the kids and go to the embassy and ask your questions, but getting the visa at IST last month took us about 4 minutes. We did sit towards the front of our plane ( just plain ol economy class).
You probably know this, but go to the visa line FIRST with 20 dollars US in hand per person, it is just a sticker they put in the passport and voila it is done. We saw folks that were sent back from passport control to get the sticker because they did not stop first.