What type of Non-O visa should I apply for as a dependent of my husband working in Thailand?

you need documentation from his employer stating it's okay that you get a Non-O visa based on "trailing spouse" and the kids get them based on being dependents.

You can apply for 90 day single entry Non-O visas in the thai consulate of the country you're in BEFORE you wing your way here OR you can all come in and then apply for the 90 day Non-O visas at the immigration office where you live.

I'd say get the Non-O's before you come here if you can.

Once you get the 90 day Non-O visas you then wait until you have 30 days or less left and apply for a matching extension that your husband has at the immigration office