Can I obtain a tourist visa for myself and my husband while my children use visa exemption, and what about potential overstaying consequences?

If you are from a country that gives you 30 day visa exemption all of you could come in this way. You get 30 days again on each subsequent entry.

I would avoid a plan of overstay. With or without visa, immigration officers (IO) can deny entry for any reason. Despite what many may think or say, they don't really need to reason to deny you entry. A plan of overstay suggestions you don't care or respect the rules. Immigration officers love the rules. This may put you in a bad situation.

To the point about IO authority to deny you entry without a reason, maybe they can, maybe they can't, but do you really want the hassle

Best to follow the rules.