The periodic wet brown stools that most healthy young chickens have will usually be red and watery if a chick has cocci. For the most part Corid is a good preventative treatment for coccidiosis, but it has the side effect of stripping vitamin B from chicks.
I think you were smart to refrigerate a dead chick because whatever your chicks have, it seems to be contagious. Have you considered getting in touch with your county extension service and seeing if they can in turn put you in touch with the county poultry expert? That person may be able to readily identify the problem and also may be able to perform an analysis on the dead chick.