It's a frustrating situation when you think you have travel plans settled and then they blow up like that. Take a deep breath, rebook your kids on acceptable flights, and request your refund and credit.
It might also be a good time to sit down as a family and review your flight contact information in all your frequent flyer accounts so that you can avoid this kind of snafu in the future. If your wife is to be the contact for family travel, she can make sure her contact info is in all the kids' accounts.