Anyway, on a more serious note

, we probably don't have billions of years, out of the 4.5 billion we think the sun 'has left', the sun will increase it's energy output over time (assuming we don't fiddle with it?). I believe the amount of solar radiation that will cause Earth to go into a runaway greenhouse effect, like that on Venus, is going to be reached in 400-500 million years time. So that's really one of the first real 'natural' deadlines!
Mars may be quite a nice spot by that time with the extra heat. And a billion years later when Mars shrivels, perhaps relocate to Titan.
Anyway, I think your being a bit too panicky! Homo Sapiens have, as far as the evidence we've found only been around for 200,000 years, and in the mere blip that was the past 5000 years or so we stumbled upon civilisation and have since found out a lot more stuff. (Definitely a 'hockey stick' sort of graph on innovations and knowledge!)
Of course we've still got a lot to learn to be more responsible and sane, especially about our short-term impacts on climate, environment and the plants and animals of biosphere; and how to actually behave amongst ourselves. But with that sort of progression who knows where we will be in 5000 years from now. Could well be at the stars - or we could be a few tribes struggling in very hostile climatic conditions on a ruined earth, I suppose.
Personally I think going for a full-scale Dyson Swarm, at least at some point, makes more sense. Loads of space, loads of life, and loads of minds that could be employed for any grand project that might take our descendants fancy: garden ships, generational ships, warp drive etc...Maybe in a few million years Sol will be feeding a vast cloud of O'Neil orbitals that have a billion times the land area of old Earth, with a billion times more sentient beings than today...
...although who these sentient beings will be, I don't know. I read somewhere that the average mammalian species lasts 3 million years before going extinct, perhaps if long-living, the article suggested that you might get ~10 million years. Of course said species may not be around but their descendants could be well and happy. So whatever species come about from our evolution: both natural and artificial would be in this future....