Answer: It sounds as if your plant is suffering from rot. Rot is actually a symptom, rather than a disease, with tissue becoming soft, black and decayed. It usually sets in when plants are suffering from stress due to cultural needs, such as overwatering or low light. The stressed plant is weakened and bacteria or fungi move in. (Sometimes water can accumulate at the base of pots and roots rot.)
However, it's odd that the other plant in the same pot is doing well (I'm assuming that it's also a dracaena.) Sometimes, one plant is just healthier than another. Also, sometimes in transplanting if there is disease/bacteria on a tool, pot, etc. it can be transmitted. It's a good idea to sterilize pots and tools between work sessions.
If the entire stem of the affected plant is "squishy," I don't think you can save it. You'd be better off getting it out of the pot so it won't transmit to the other. Otherwise you can cut off the affected portion, dust the remainder with sulfur, and see what happens.
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