It is a grafted cactus  some form of epi on the bottom & a gymnocalcium on the top. You can graft the top cactus onto some type of epiphyllum. You might try letting the bottom epi harden off & root in sand or peat  depending on the severity of the damage. These grafts are often rooted in peat but eventually need to get out of the peat. Otherwise they tend to de-graft &/or the bottom tends to rot. I have been able to get them to bloom in sand. I have also seen very mature specimens in full sun in cactus soil. This is not the greatest environment for the epi base, but the plant was probably 10 years old & plump, which is a pretty good track record for this plant. The verigates lose their color in full sun, but the coloration tends to come back when placed in shade.

I took the little guy, did a tiny dab of alcohol directly onto the black surface, and covered the area with cinnamon. I cut into the big guy to clear out the black spots, covered it in cinnamon and left it overnight. This morning, the area above the damaged skin was softer than the rest of it, so I cut the top off an inch or two above that. There were still some tiny darkish spots, so I cut an inch above that. Those spots were still in there, but I could trace them to the aerial roots, so I covered all cut surfaces in cinnamon and am not going to touch them again for a few days. I have them on top of my stove, which is shadowed in a room that gets bright but not direct light. Current plan in a few days is to clear the cinnamon off and put the pieces in small pots with the grittier soil mix and then continue not to touch them. x-D I'm open to suggestions, and I'll follow up when I know how it goes. Thanks for all the help, and I'll have another question posted soon. x-D

When I look at the photo - maybe the soil was too heavy/organic. Succulent plants need fast draining soil, containing a lot of inorganic fraction (grit, river sand, perlite or similar materials).

Rachel, it's very hard for me to make an assessment when I am not physically there to examine your plant. I don't quite understand what you mean by soft but not overly. I see that your last picture you had lost a couple of stems, so perhaps if its really rot setting in the base, you need to catch where is the rot happening at. If it's right under all the stems towards the roots, you are better of cutting off as much rot/brown portion as possible until you see healthy green stems, and try to root the cutting as best as you can. At this point, you can try to snap 1 or 2 healthy cuttings at the tip end towards the base. Always let the cuttings dry up and callous before their to root them in new soil. And also make sure when you snap them apart, the stem must be close as to the root base as possible, if you can snap some roots with the stem even better.