Q: If I have an Evil Magic Ring activated (face-out in my hand) and someone Trades Hands with me, does the Evil go to the person who gets my hand, too?

A: Yes. Since Evil would, in this case, stay with the Magic Ring, held face-out in your hand, it would go to the person who gets your hand. At that point, the Magic Ring would be turned inward, detaching the Evil, and depositing it on the table in front of its new owner, where it will attach to a different available Keeper.