Your Overwatered Succulent Can Still Be Saved by Cutting
A succulent whose roots and lower stem have rotted is still salvageable, but only if the roots are written off completely. Take the plant out of the soil, cut the stem above the rot into clean, firm, evenly colored tissue, let that wound callus in dry moving air for several days until it is hard to the touch, then stand the cutting on gritty mix and leave it bone dry until roots form. The old soil, the old pot, and the mushy base are finished. This is rescue, not propagation. The plant already in your hands is the only reason to make the cut, and it only works if you finish the cut.

I've done this at ten at night on a kitchen counter with a paring knife and a paper towel, and I've done it badly. The badly version is worth describing, because the mistake I made is the one almost everybody makes: I cut too low.
Find the clean line
Here's the moment. You've pulled the plant because the lower leaves went soft and translucent and then dark, and the whole thing lifted out of the pot with no resistance. There's a smell. Sour, wet basement, not quite dirt. The stem at soil level is a brown sheath, and when you squeeze it, it gives like a grape. Under the sheath there's sometimes a pale core that looks almost healthy, and that core is what tricks people. It isn't healthy. It's the part of the stem that hasn't finished dying yet.
Slice an inch above where the mush ends, with a clean blade. I wipe mine with rubbing alcohol and let it dry, because alcohol on an open cut isn't doing the plant any favors. Look at the face of the cut. What you want is uniform: pale green or cream, no dark ring circling the core, no brown streak running up one side. A dark ring means the rot is still traveling. Slice again, above it. And again, until the cross-section is clean all the way across.
Then cut one more time, an extra half inch above the clean face, and that extra slice is the one that matters. Tissue at the edge of a rot front can look fine and still be colonized. That margin is what buys you the plant.
Sometimes you run out. You slice and slice and the stem is dark all the way to the crown. A plant can be all rot, and the honest answer then is that there's nothing to save. Better to find that out at the counter than three weeks into a cutting that was doomed when it left your hand.
The dry wait
The cut end is an open wound, and it has to close before it touches anything. Dry, airy spot, out of direct sun, not in a closed cabinet, not on a saucer with water in it. Three days minimum; in a humid summer house, closer to a week. Then touch it. You're feeling for hard and dry and slightly corky or domed, like the end of a dry twig. If it's still cool or tacky, give it more time. There's no penalty for waiting longer.
The plant will shrivel while it sits there, and it may drop a leaf or two. That's water leaving, and it's supposed to happen. Everything in you will want to water it as a response. Don't.
Potting a cutting that has no roots
Small pot, barely wider than the stem. A two-inch terracotta or a rinsed nursery pot. Gritty mix, the kind that's mostly pumice or perlite. Dry. Push the callused end in about an inch, or rest it on the surface with a couple of stones holding it upright in bright shade. The old soil goes in the trash, and the old pot gets washed before it's used for anything else, because that soil is where this started.
No water. Not a spritz, not a bottom soak. A cutting with no roots sitting in damp mix is a cutting rotting from the other end.
Check for roots by gripping the stem and giving a slow upward tug. Before roots, it slides. After, it resists, and there'll be small white nubs at the base. A new leaf pushing from the center of the rosette is the other signal. Once you have one or both, water lightly around the base, not a soak, and let it go completely dry again before the next one. That's the whole transition. People lose this stage by going back to normal watering the first day they see roots.
Sulfur does one thing
Sulfur dusting on a fresh cut is a traditional physical barrier. It keeps the surface dry, it makes the wound less attractive to fungus gnats and surface mold, and I keep a small container of it on the shelf. It does not kill rot inside the tissue. If there's still rot in the cutting, dusting it just puts a yellow hat on a dying plant. Dust a clean cut. Don't use it to skip the extra slice, and don't use it as permission to water.
The leaves you pulled off
Lay them on dry mix in a bright, shaded spot. A leaf needs a clean heel. If it came away whole with a tidy nub at the base, it has a chance. If it tore, or the base is dark and wet, toss it. Leaves off a rotten stem are slow, months, and a leaf whose base sat in rot may carry rot into whatever grows from it. That's a consolation prize, not the rescue. Don't let it pull your attention off the top.
When cutting isn't the answer
Not every overwatered succulent needs surgery. If the stem is firm and the leaves are soft and translucent and the soil is soaked, dry it out, repot into grit, and wait a couple of weeks. The cut is for mushy bases and dark stems. It's easy to panic and behead a plant that only needed to dry out, and that costs it weeks it didn't need to lose.
And the most common way a rescue fails isn't the cut. It's watering three weeks later because the cutting looks pitiful. A shriveled cutting on dry mix is doing its job. A watered one on wet mix is the second funeral.
Educational content only: check with a qualified horticulturist, licensed pesticide applicator, or your local extension office before acting on anything specific.