Probably a similar field to OTL. Most were running because the door was open and they wanted the job.
Without Iraq as the elephant in the room, the primary results can change. Domestic issues are at the forefront so Gephardt, Dean, and Edwards are stronger. Kerry won in part because Democrats wanted to relitigate the National Guard issue, and that would have been less salient in peacetime. (I'm assuming no Iraq 2 Electric Boogaloo with no Afghanistan, which isn't guaranteed.)