You must cut the rot off of the root or it will spread. See my thread about dealing with rot.
Basically, remove, wash and dry the plant. To speed the process, set the plant wet from washing on an absorbent cloth or paper towel. Cut the rot off of the root, dividing the plant if necessary. Be sure to remove all rotting leaves. You must remove all rotted pieces from the plant or the rot may spread to healthy parts. Dry any remaining healthy parts of the plant for at least a day or possibly a few days if there was a lot of rot and you are not sure that you got all of it. Put them in a cool garage or other similar place, out of rain, snow, etc.
Once dried, plant and mist only the plant after a few days. Do not water when planting. Watch carefully for more rot and protect from rain and snow for a few weeks if it is winter.
I've concluded that heuff's in general probably need more water during cold temps than sempervivum. That is a generalization as many of both familys have differing sensitivity to moisture.
I have not seen "spongy roots" so not sure what that may imply.
I am a fan of dividing heuff's if they are large and healthy, but that is a judgment call. If it is during a stressful time, you do have to be careful with rot. In all cases of dividing, you must let them callus and dry after dividing and before planting, then be stingy with the water, unless the plant is starting to dry up (the outer leaves will tell you). If the plant is growing rapidly, then its ok to water but I like to wait until the plant starts say "Please, sir, can I have some more (water)!"
