What are you teaching your children? When you have a tough time, you just send it away? My greatest fear for you, especially if the kids love the dog, you will be setting up a problem that will cause them resentment, and many other emotional problems.
Since you appear to be the primary care giver of the dog, I'd call a family meeting. Tell them that something needs to be different. YOU are going to take the dog to obedience school. ( there are so many out there, call your vet they know the ones that come to your home or the places to take them) They need to supportive. and do what you tell them when it comes to dog management. YOu will need to do some homework in between classes. The kids can also learn the commands
The problem is NOT the dog. He is just doing what comes naturally. He just doesn't know what he is suppose to do.
The training will also be beneficial for your children. When they are in a different situation, they will know that they can do and how to be the "alpha" dog over anyone they meet. Dogs want discipline. We need to know how they understand it.
Think long and hard about this decicison because the long term effect could make a very strong impact on EVERYONE, including the dog. IF he knew what proper behavior was, I'm sure he'd do it.
Sorry, if you didn't like my answer, You asked. I'm just so tired of people bailing and giving up. Finding solutions to the problem will be a learning experience for the children as well, they will respect animals and they will have fond memories in later years of the family dog.