Lindsay, I had a couple start to look like that after we had the drenching from Debby. I did some cutting on the soft or discolored parts, and just doused the rest including the cuts with cinnamon. Seems to have worked, they all recovered except the one little Paph. I didn't notice him in trouble until too late. One thing about cinnamon, it most likely can't hurt!
I kept all the affected ones quarantined in the shade and under the roof overhang where they couldn't get rained on any more, until there was no further sign of plague. Just misted the leaves, being careful not to wet the affected areas, or to splash any water around.
From your pics, I'd take a sharp serrated knife and cut out all those stems with the dark discolorations, then as I said, douse the whole pot with cinnamon. Until further expert help advises otherwise, that is.
