Q: How should one deal with rumors (especially false ones) about oneself? Should one spend her time chasing them down and trying to explain their false nature, or ignore them? Is there some other alternative?
A: The first task is psychological: treat the rumors for what they are; that is, as falsehoods. Perception is not reality. You know what's true, and others important or close to you know what's true. Only the truth comprises reality, not what anybody says. People who spread rumors, or even engage in gossip, are dabbling in non-reality and enjoying it. It's no different than dabbling in drug or alcohol abuse, although it doesn't involve a chemical effect on the brain. It does, however, warp the soul.
When dealing with a rumor, the general policy should be not to comment on it or address it unless there's some compelling reason to do so. There might be an emotional tug to address the rumor, but you have to stop and ask yourself, "Why do so?" If the rumor consists of an outright falsehood--as opposed to the opinion of someone who does not matter to you--and if the falsehood is reaching the ears of people who matter to you (either professionally or personally), then it certainly makes sense to give these important people the "heads up." But for the most part rumors should not be given the status of attention-meriting material at all. The old saying that "gossip is the hobgoblin of little minds" couldn't be more true. People who spread or engage in rumors and gossip matter less than just about anyone who travels this planet. Grant them the status they deserve.