To answer your title question, there isn't a way to cope with careers that become extinct due to technological progress. You just simply choose something that's in demand, and hope you chose right.
What most people don't realize is why everything works the way it does, and how we got to where we are. As a result, they are doomed to repeat the failures of the past as cliche as that sounds, and the most recent and latest trend is and has been in stripping you and everyone else of their agency and voice.
We receive many of our benefits through the division of labor. It is best described in a book from 1778 by Adam Smith called the Wealth of Nations. There have been follow-up books such as Landis which covered why some civilizations thrive while others stagnate with climate being a main factor, specifically how tropical climates make animal husbandry impossible and settlement at a high cost of mortality.
Our system worked so long as our leaders listened and acted on what we said, that hasn't happened in a long time. Today they can't differentiate the voice of bots on social media from actual people. They have written laws binding themselves to prevent any change, or only benefited special interests who got them chosen to be on the ballot with money.
Worse, global communism (or socialism) however you want to call it is intentionally attacking the moorings of our society through subversion, we aren't taught about that in school and so don't recognize it.
Most people can't even properly define what socialism is, know its failings (which are not straightforward) and recognize the lies. Those same people then go and push socialist policies as if its the answer.
Worse, there has been a 50 year trend of not having a funding plan for things, and choosing to steal via ponzi from the future with creative financial engineering, corruption, fraud, and collusion.
We are coming upon a criticality point where everything will break, because machines can do most jobs better, and cheaper than people in almost all important factors.
We are not that far off from having machines that can do everything a human can, and do it better and more accurately.
Our leaders have failed in almost all respects that matter with regard to planning for the future, instead choosing to subvert, divide, and limit agency by political cycle, through many various means (media etc) while pretending to do their jobs in a way that nothing really gets done.
Eventually someone will come up with a way to break the existing theory of computation, and then machines can completely replace human productivity and shortly after that everything will break.
People say this is impossible, citing that theory, I'm not going to go into why you shouldn't use a definition to define itself. Put simply, its possible because we exist, and are capable.
So how far do you want to go down the rabbit hole?
What does your philosophy and ethics say you should do? As a worker do you support this path. If you work for them you are supporting them. Do you support companies that replace their jobs with AI or continually offload costs onto their customers, reduce their agency, de-amplify your voice, and enslave you through debt?
Businesses will first replace the low hanging middle class jobs that cost them the most, then the lower paying jobs. In a perfect corporatocracy you would have only leaders that own machines, and some technicians to keep the machines running, and no workers because labor is the most expensive cost.
What can you do about it? Nothing really, not directly.
The chance to actually do anything about it was in the 90s, and that's where the last generation utterly failed and broke the generational social contract.
We have, as the economists call it, dead men ruling, and they have made it practically impossible to right the course.
The only thing you can do is learn to become completely self-sufficient, develop a community with specialized skills and resources, and that might offer a slim chance at a future for your children.
Things will get much worse. We can't avoid what's coming, but we can work to ensure its a beginning and not an end for our children.