Cut any of the rotted branches back to a point where you find clean healthy growth. If the rot started in the roots, you may only be able to salvage a lot of cuttings, however if the rot is at the tips, you can cut it back to healthy wood, and your plant will come back from the rootstock. I hope that makes sense to you. My crown of thorns has developed some problems over the past several months. Now the plant is not very full, the leaves get brown and yellow spots on them, proceed to dry up, curl over, turn yellow and fall off with alarming regularity and speed.
Several leaves fall off every day.
The plant isn't nearly as full as it once was and the spines on the stem are brittle and break when you touch them instead of poking you. Please advise - you're my only chance. Since the downhill slide of your Crown of Thorns began about the time of your move, my first guess would be that the heating is different in your new home. Even though a Crown of Thorns is a succulent, it has limited water storing abilities, and has to rely on moisture it takes in through its leaves. I would try misting your plant daily, and maybe increase the amount of water you give it each time you water it.