In general, jades resent change (showing it by dropping leaves and sometimes minor branches), and you know this one has been moved and overwatered. You said exactly the right thing about cutting it back beyond the rot. As for the drooping branches, it could be caused by getting significantly less sun than it used to get, the overwatering, or most likely, both. When any plant is overwatered the roots rot. After the roots rot, the plant can't get enough water up into the trunk/stems (which got too much before the roots went and started to rot, too). At this point, parts show signs of overwatering and other parts of underwatering. The best you can do is get it into as much sun as you can in a place where it can stay. Cut it back as far as you need to, and don't water it for a month or more. You will certainly need to do some judicious pruning over the next few years as it recovers and grows from where-ever, but jades are resilliant. It should make it ok.