@Stephen ***********
I don't want to make a mess, but just a thought.
While your children would be fine to just enter/exit Thailand when they are overstaying (only just get an overstay stamp on their passport), if you are seeing them enroll in school in Thailand, you may want them to be here legally.
If this is the case, have your wife check with the Thai embassy in Korea if they can get Non-O dependant on you like yours.
Or, if they have dual passports, make sure to let them enter on visa-exempt (not on visa waiver/bilateral agreement with Korean passport even it gives you 90days), because visa-exempt entry stamp can be converted to Non-ED for example, but entry on visa-waiver cannot.