the whigs did not disappear overnight. They fractured into the the consitutional union party and the gop. The dems fractured into the northern dem and sountern dems. Each party ran its candidate. the GOp formed from the ashes of the whigs.
The similarities to today are profound. the Tea party is what the GOP was in 1856 or the Whig holdouts whihc remains to be seen but a fracture has occurred it will not end until one or the other is defeated.
You have your facts right, but I think that the interpretation is a bit off. If the Republican Party were to go the way of the Whigs, then a third Party would need to be formed and to nominate a candidate capable of winning before the Republicans nominate their candidate. If the third party is a spawn of the sore losers of a nomination fight, they lose.