Have not been to Branson, so no help here on CGs. I usually just serch "Campgrounds in the _____ area" and go from there, but feel pretty sure you'll get more solid info that I can give. Another option would be to call the Visitor's Bureau and get something emailed to you - you're not going to have time to wait for their usual travel brochures. For sure you need to call whatever campground and confirm their W/E/S status.
My only concern with your timing is that the weather in that area is positively dreadful....you may have a problem finding something open. As far as I know, most MO state parks do not have water to the site, so you may need to plan accordingly.
Hope you can find something great and have a really good time.