I'm sorry ~ I wish I had better news for all you spa lovers. It would be an excellent way to save money during the festival, as the motels and minbak can be more than double their regular rates. My best advice for now would be to get yourself a minbak room. After all the sun and sand you could check out this sauna for a little rinse-down then go out walking around/drinking/eating until dawn as others will. Anyway, with a minbak or motel room, you'll have somewhere to leave your bag.
I'll be attending the Boryeong mud festival on the second weekend (July 17th - 18th) so at that time I'll update this post with more information with details of the Haesu Sauna. Until then, if you have any information about good motels or minbak, and especially about a sauna or jjimjilbang, please comment on this post.