I haven't done any calculations on this, but I believe that the Earth and the Moon should be considered one body of mass when calculating its solar orbit.
So if the Moon was perhaps vaporized, e.g. with a huge nuclear device, perhaps the loss of mass could be sufficient for the Earth to escape the Sun's gravity. But I have no idea how long would it take?
Or maybe, if the idea is sound, you could first get the Moon to escape Earths orbit in some way. Whatever amount of energy would be needed to get the Earth to escape the Sun, only 1/6th would be needed to get the Moon to escape the Earth. And if such an event was the collision with a very large stellar body, the collision itself wouldn't itself kill off humanity.
p.s. Completely unrelated to the actual answer (I don't have enough rep to add comments), Arthur C. Clarke's book The Songs of Distant Earth is similar to your plot description, i.e. humanity trying to create a future for itself as the Sun is about to go supernova using sub-FTL technology