Hello rootbeer,
When I've taken cuttings from these plants I tend to take them with an aerial root attached.
Thats the brown root that comes from the stems, these roots grow downwards as if they are trying to find water & soil.
I've had sucess doing this way after a few weeks. From what you say it sounds like your cutting is starting to rot,but maybe just wait & see if roots develop.
hope this helps
D.G Hessayon- "House plant expert" gives excellent advice on this,you can pick up his books in charity shops.