I would guess the main thing it would take for a major war to break out in Korea would be for North Korea to do a bit better in the skirmishes, leading them to conclude that "the decadent and corrupt capitalist imperialst regime is ready to fall" and attack.
When the attack happens is fairly important, if it comes after the war has become immensely unpopular then the US might use it as an excuse to gracefully leave Vietnam; if it comes before the public has turned against the army then an attack probably serves to make Vietnam more popular, and might even prompt the US to go into North Vietnam rather than staying in the South.
Managing a two-front war would probably not be as difficult, as the Korean war would be a much more conventional conflict where US tanks and other heavy weapons that were not suited to Vietnam would come into play. Barring intervention from China or the Soviet Union North Korea would lose a conventional war quite quickly, this might prompt North Korean leadership to attempt a guerilla compaign once it becomes clear that defeat is inevitable. The US fighting two counter-insurgency campaigns in Asia would be a difficult thing, and the initial popularity boost granted by US interests being attacked would probably fade before long (look how popular Iraq is now).