John Edwards' scandals won't be an issue because Vice President John Edwards would never have even *met* Rielle Hunter. So he's still in good standing and, regardless of whether or not Kerry wins or loses in 2008, he's a clear frontrunner for the Democratic presidential nomination in 2012.
Kerry will likely do an earlier 'surge' than Bush did, owing to his own campaign rhetoric. Which will probably help him out with securing his re-election campaign, even though it'll probably hurt him a bit with liberals, who'll probably see Kerry as most of them see Obama now. Otherwise, foreign policy will probably take the same route that it has under Obama, stressing reconciliation and mending American relations with Europe and Russia. Kerry will probably pay lip service to cracking down on foreign trade manipulators much in the same fashion that Obama has done, and I still see most of Bush's free trade agreements going through. A Kerry administration will also probably reorient the U.S. toward finding bin Laden and he might end up dead much earlier than IOTL.
On domestic policy, Kerry is going to have to deal with Republican control of both chambers of Congress, which means that he'll have a situation similar to what Obama has now. I can see him working a bit better with them though, largely owing to that Congress being significantly less right-wing than OTL's 112th Congress. Remember that the economy wasn't *great* in 2004, so I can see a President Kerry working with Congress to pass a small stimulus package that would help spur it on while also working toward reducing the deficit. Of course, most likely nothing will get done about the banking crisis, which will blowup in the year before Kerry seeks re-election. I assume the Democrats will make gains in the 2006 midterms (at least in the House) but still not take full control of the Congress at any point in Kerry's presidency.
By 2008, the Democrats will start to fall in popularity as the economic crisis hits hard and Kerry becomes increasingly unpopular. Republicans nominate McCain (with VP candidate Tim Pawlenty) who beats Kerry in a closer election that OTL, but still not a very close. McCain takes office in 2009 with a Republican majority in both chambers and the intent to stimulate the economy and deal with the banking crisis with tax cuts and deregulatory measures, though he does support the reinstitution of the Glass-Steagall 'Wall of Separation' in banking.
By 2010, 11-12% unemployment allows the Democrats to win control of both houses of Congress. John Edwards leads the pack of potential Democratic Presidential nominees, while also facing down opposition from Senator Hillary Clinton. In the end, progressive activists manage to nominate Edwards for the Presidency. McCain announces that he will not seek another term, allowing Mitt Romney to claim the nomination. In a blowout for the Democrats, John Edwards and Vice Presidential nominee Barack Obama take the White House, increasing their majorities in both chambers of Congress.