a visa run means you go apply for a visa via a Thai embassy. Check if your passport country is ok for visa-exempt ( no visa) entry of 60 days and do a border bounce.
so you're now in on visa exempt? So no visa, but can stay 60 days and can extend by 30. That should give you time to get DTV corrected if it was their mistake.
Are you coming on in a visa? A visa exempt? Fyi as an American you don't get visa on arrival. The immigration can ask for evidence of cash so maybe that's why airline asked.