It sounds like they have root rot. The plants may be saved by cuttings. But the main plants are goners.

Thank you, mrimpatiens. The plants seem firmly rooted, so can the problem still be root rot? Also, when I've tried to root cuttings from this plant in water, they've wilted and died. Any suggestions?

Hmmm root rot was my guess. I always start my cuttings in moist perlite and cover it with plastic wrap. Have you checked the sides of the pot? If they are dark its more than likly rot, if not... I have no idea. Some roots could still be holding it in the soil.

Well, the sides of the pot don't seem to be black as far down as I could check by pushing back the soil next to the pot. I also didn't mention that leaves are turning black, just a few each day - not en mass.
But I will try rooting them in pearlite. Just pearlite and water covered with plastic? Thanks!