I think I posted this under the wrong forum so I am reposting on the Cacti and Succulent forum. Sorry for the confusion...

Use a clean knife to take off the mushy roots. You will need to cut off the branch ends as well. Cut to firm plant material. Then dust the ends with a fungicide and let dry a few days. I had some rot on the caudex of mine that I had to cut out. I used cinnamon on it as a fungicide and it worked very nicely. You can repot once they roots have dried a little. Repot in well draining cactus mix with some extra perlite mixed in. Water sparingly at first and when it shows new growth treat as usual.

I see no mention of perhaps, if your plant is "indoors" or "outdoors", since in your climate many tropicals are left outdoors perhaps through the winter months, and I do recall a severe cold snap or ice storm this past winter no too far from you perhaps. Could be cold damage, if it was expsoed to very cold temperatures. I'm hardly an expert however, but mushy tips cold be cold damage. They are even more sensitive to overwatering of course under cool growing conditions. I guess it bite the bullet though regarding removing abd trimming away the rotten areas and allowing them to grow on the dry side afterwards til they've recovered.