I've never really understood the pessimism of the dying sun trope in SF. If humans are still around in billions of years then our descendants will either have spread throughout the galaxy, or will have the technology to either tweak the Sun to not go nova, or to move the Earth out of harm's way. Seems like a major failure of the imagination. (Also: the idea that we need to find other planets to live on if the Sun goe…
For one, why would we colonize the galaxy? We probably can't beat lightspeed. It'll take generations to get anywhere, plus a huge upfront investment, and for what? We can't send resources back, so it is not an investment. Terraforming would be extremely challenging, extremely slow, and would (if we were very, very lucky) still just get us a worse version of Earth. It just isn't a rational choice.
"Tweaking the sun" is probably unfeasible. We could crash our entire planet into the sun and it'd have little effect. The sun is really really really big and we are really really really small. Similarly, adjusting Earth's orbit would require a colossal amount of energy and reaction mass which we just don't have, and probably never will.
There's no reason to assume that humanity will inevitably advance through some exponential techno-apotheosis such that we'll eventually be able to do anything we want. Instead, it's much more sensible to assume technological growth will follow a sigmoid curve. Technological advancement has a ceiling determined by our resources and abilities, and that ceiling is almost certainly too low to accommodate a galactic civilization of stellar engineers.