i would not be surprised if humans died off within a couple centuries after that.
I would. If one or more isolated populations managed to survive more than a couple of generations after the event, I think it's highly likely that they'd continue to survive indefinitely. The worst of the changes would be past, and they'd clearly have learned how to survive in the new environment, else they'd have died sooner.
Human intelligence makes us highly adaptable, as evidenced by the extraordinary diversity of environments in which we live, and lived even before the advent of modern technology. Humans who lack the necessary knowledge of how to survive in a particular environment are at severe risk of death any place on the planet, but if they manage to survive for even a year or two, odds are that they'll have learned enough to be able to extend that time almost indefinitely.