That is so wrong. I am in shock reading that stuff. It seems really crazy. Recently you might have heard that the Korean Govt have been rushing to implement a whole bunch of "well intentioned but poorly planned/ thought-out/ advised new laws for visas". I suspect you have been caught up in this. Our uni has lost several potential and serving teachers over this, and many Unis and so on are pretty furious at the way it's been handled. They are getting short-staffed and made to loose face. But, to the point of whom might be able to help you:
1-I'd be contacting the Dept of Immigration first up, because it is really their call on visa approvals. Dept of Justice wouldn't really be involved for minor traffic infringemments...that sounds like someone passing the buck.
2-You could try contacting a lawyer who specializes in visa law. An internet search should get you this pretty quick. They may cost, but usually they can at least answer a call/ e-mail for free and say whether there is anything they can do or suggest.
3-If you wanted you could contact the Dept of Justice, though their web-site seems to be down at the moment.
If I learn of anything else that might help you out, I'll pass it along. Best wishes and good luck to you and your family Bryan!