This is just my opinion, but...
It doesn's sound like you need a dog. You sound like you are already busy and, ultimately, the dog will be your responsibility, too, and not just your children's responsibility. Honestly, how much more do you think you can take on and not go crazy?
I have two kids. I also have three dogs. My kids are still small, so they are not involved in many activities yet. I am home all day, so I can spend a lot of time working with the dogs. And that is exactly what it is- work! They have to have attention and exercise, and they need cuddle time, too, just like the kids do. Not to mention training! My dogs are very active (two pit bulls and a pit bull mix) and have to be kept occupied. I love my dogs and wouldn't trade them for anything, but they are a huge financial, emotional, and time investment!
Maybe you should wait until your kids are not so busy to introduce them, or yourself, to any more responsibilities. Just have a talk with them and explain why they cannot yet have a dog...they are old enough to understand you.
If you do decide to get a dog, try to get one that is more fitted to your lifestyle. Let me warn you, though, it will be hard for you to do if you have a very hectic schedule. Virtually all dogs need a lot of attention. Cats (really) are the less-needy animals.
Do some research on the different breeds...ask a vet...talk to the humane society. Really think about what you expect and the ways you can accomodate whatever animal you decide to take into your home (ie are they indoors or outdoors, how much time can you invest in training etc etc etc). Educated pet owners make the best pet owners.