They really need to...
Stop worrying about what Twatter does, and have the people from the media who actually believed it was genuine taken away by people in white coats.
When it comes to simple facts like where a person's web site is the media should be a trusted source. This means that the media must in turn ensure that it gets its information for whether it is genuine from a trusted source. The twatter feed is not a trusted source.
In summary, the reason this hoax got so big isn't because of Twatter... it's because of the twats at the media who didn't voice any skepticism at all. Even a "flimsy arse covering statement" would give people something to think about.
Why haven't I seen anyone calling this certain web 2.0 microblog service Twatter before? Surely it's pretty obvious?