Cut some long strips of cloth, I prefer cotton, as old bed sheets, and thread the strips through the roots and hang the plant in the warmest place you can find. Let it just stay there for weeks! It's had enough water for a while.
The root fell off? Then it's healing! That's good. I wouldn't cut any more right now. If you want, try a little slice in the soft spot, just 1/4 of an inch, to see what's inside. If it's mushy, and slimy/stinky, then go ahead and operate. You needn't chop all the way to firm white flesh, just get close and then rub with paper towels to dry off as much as you can. Then maybe even set a hair dryer on low, and put it far enough away that the plant is warmed, but not heated to being hot. For an hour or so. I've done this, and had success.
Once a plant is over watered, and rain is water (!) the tiny feeder roots die, then the small roots die and start rotting, and the plant isn't using the water that's in the pot, and that is rotting around what's left of the struggling plant's caudex and roots. I'm glad you did take it out of the pot. Fresh potting mix will be really good for the plant, once you repot. Don't pack the medium into the pot when you do. It should be just barely moist, and holding the plant, shake the mix around the roots. Then don't water for at least a week, so the plant can start new feeder roots. Whew, lotta typing!
