Actually, they go into it early on in the manga, when Emilico is scared that Kate got disintegrated in that vacuum room (I don’t remember exactly what it’s called). Kate explains that even though Shadows emit soot, they are not themselves made of soot. So, they can take baths and stuff.
In fact, that’s part of the strenuous routine they and their living dolls need to do to keep things clean given all the soot they emit. They also change their clothes multiple times a day, I believe.