@Brian *******
you can enter Thailand...as family. Using their US passports. All entering visa exempt...following which you should have the children obtain their Thai ID cards and Thai Passports.
You can apply for a Non O Visa as father of Thai Child. It will require deposit of 400,000 Thai baht in a Thai bank account in your name only.
When the children leave Thailand they will use their US Passport to depart.
And Thai passport in future entries.
As its night in Thailand ...lets await further reply from our Admin/Mod team in Thailand in the morning.