That looks like the first above ground signs of a potentially major problem. I know that for some reason these plants are now pretty popular as indoor house plants, but they really are not suited for that.
If you are in the NE US it is a really bad time for that plant to get this, but also not an unexpected situation.
What to try and do with the plant: try and bare root it - take it out of the soil and wash as much of the soil off. Hopefully these are two separate cuttings in the same pot and the other is still in OK shape. If this is the case you should let the unaffected plant dry out in a warm dry place for a few days ad plant it in some cactus and succulent soil amended with pumice if you can get it, but coarse gravel or perlite will work too.
Then find the warmest sunniest place you have your house and put the plant there. Water it at most once a month. Right now you want that plant to survive but not grow - since the light conditions are poor for this plant where you are at the moment any growth will likely be etiolated (weak spindly growth compared to the natural growth form).
The other plant with the rot, probably has major issues in its roots and really the only thing to try would be to cut right above the rot to see if the main tissue in the plant above that spot is healthy looking if it is not you keep cutting until you hit healthy tissue. Then let it sit and dry out for a good week. Again in a warm and dry spot. Once that is done sit it in some of the same soil mix as you made for the other plant and do not water it. Put it in a warm, well lit place. It may re-root and once it does you can water it sparingly.
I am worried that the plant with the rot will have hard time making it given that you are going to winter. However, if you can get a clean cut and then keep what is left dry you might get it through winter and once it warms up in the spring it may have a chance.
If you got it about a month ago there is a chance the plant already had a problem going when you bought it, but that will be hard to be sure of. Alternative reasons why it developed this without you water it is that it may just have been in too cool a spot, that may also have been too humid.
The narrower growth on the one plant is a sign of etiolation: too little light and warmth. That is what they tend to do when grown indoors regardless of the other conditions.
Just to put it in perspective these plants grow outside here in unshaded full sun and generally are very happy with just a little extra water. They are made for crazy desert heat and low water conditions. Growing them inside in an area that is relatively humid is about as far from its ideal growing conditions as it can get. It is not impossible to get that plant to survive in such conditions, but for it to thrive is a whole other matter.
Good luck!
