Instead of coming up with ideas that focus on "saving" humans à la getting them out of the solar system et cetera, why not simply focus on leaving behind a collective (human) legacy?

What would keep the world together in such a scenario, would be a collective feeling of hope, and not a selective one, I.e: The world works towards the "goal" of getting a very select club of people off this rock, while leaving the rest to their inevitable demise.

What could a collective effort look like?

Well, I'd say we prepare some new and sturdy shuttles containing the building blocks for life, our DNA as well as the DNA of many other species from the animal,- insect,- and botanical kingdom, and try to stuff as much of our knowledge about the world, our science, us humans et cetera into it as well.

And then? Well, we simply hope that either some other intelligent life form finds one of those shuttles and makes use of the information, or we hope that a shuttle crashes unto a habitable planet.

That's it. That's our last-ditch effort. Everything living on this rock will die, but at least there's a chance of continuation.