Yes, we need to leave Earth but there are two factors that must be dealt with in your scenario. The timeline on the climate disaster we have created is proving to be much faster than even the most aggressive scientific models and the size of the universe is not on a human scale.
The first issue may well mark the end of our species. We are having great difficulty in coming to terms with the crisis that we have brought on our planet. The numbers and plans, laudable as they are, that you cite now look to be only a fraction of the effects we will need to deal with and the time scale is, for at least some of these effects, more like forty years rather than eighty.
We are being brutally challenged by our inability to handle nonlinear complexity and the scale of what we must learn to manage. Just one example is the need to manage what is happening now at the planetary level. A significant part of our population cannot even comprehend what this means let alone how to do it. The regional lurch into antique fascism is a massive symptom of a chronic human disease that must be controlled for us to have any hope.
While science fiction has shaped all of us to actively desire and design our future but for most people, see above, there is no future and nothing that they can understand as a future. Unfortunately while we will be increasingly able to send out virtual agents beyond our solar system the information will be decades in returning with any line of technical development from where we are now.
We have only very vague ideas of anyway to get past Einstein’s limits. As we acclimatize ourselves to the new Quantum Mechanical universe we are beginning to discover more about how that may work but what that may mean at the human level is not yet encouraging. Quantum entanglement and the replacement of what we sense as physical objects with potentialities that are linked to actions (maybe even of sentient beings) literally makes the universe as different kind of place than we ever imagined. There are big questions there as to whether we can, in our current evolutionary form, effectively deal with that at all.
On the more practical level we aren’t going jump in our starships and flip the warp drive on an zip to even a nearby start system and then zip back home. Trashing one of the known habitable planets within our general galactic neighborhood is not really a worry as we can’t even get there. Yes, that is a terribly depressing reality and that is why we actively suppress in our entertainments. We know that our only real hope is to leave this planet.
What we can do is get very, very serious about building orbital towns and then cities. That is possible with out current technologies although incredibly difficult and dangerous. The fact that we don’t have cities in Antarctica shows how hard that will be. But there is hope there.
But the problem may be solved anyway. That problem is human mental and physical limitation. We have been and are accelerating our evolution rapidly but our evolution will not be our current biological forms. We need to get very comfortable with electromechanical and digital offspring who will move off this planet and work, comfortable, at time and distance scales for galactic expansion.
That will also be the route to saving Earth. We can’t do but we need to raise new species with greater diversity to do this. There is hope but it won’t be Homo sapiens that survives and accomplishes this.