While I would tend to believe in the theory that it is the character of Edwards that would make him eventually doing something extraordinary stupid or illegal, Edwards did have a real chance in 2004, and it is hardly believable that he would be able to have an affair with anybody during the campaign - closely watched by cameras of the media.
Who would Edwards have picked as his running mate? Kerry, Biden, Gephardt or someone else? Not sure if he could defeat Bush, but if he did, how would his presidency be like? Let's assume that he did not have an affair or involvement in unlawful practice, would he simply become a leftist populist version of Jimmy Carter, and then got crushed in 2008 by the GOP nominee, if the financial crisis broke out as in OTL? Otherwise, what agenda would he try to push forward during his first term?