Can you teach in a school with a DTV Visa, or do you need to switch to a Non-Immigrant 'B' Visa?

You can't work for any entity inside Thailand on a DTV. You would need to convert the DTV to a Non-B visa at an immigration office inside Thailand with support from the school. If the school won't support your application for a Non-B inside Thailand you won't be able to apply for it outside Thailand as you can't hold 2 visas at the same time

You can only have 1 visa at a time. DTV is work restricted. If you want to work you would have to forfeit your DTV visa and switch to a non-B visa.