I was in exactly your position two years ago. I seemed to just wake up one day and realise that climate change is real and terrifying and everybody else was acting as though everything was fine. I was in a state of panic for about four weeks solid, and I lost a stone in weight. I couldn't eat, sleep, play with my children, speak to people about anything other than climate change. Couldn't do anything.
I got some therapy and started taking anxiety medication but it felt really perverse because I wasn't wrong. It is that worrying. I thought everybody else was underreacting. I sometimes considered that I might have to kill my children at some point in the future because I'd basically brought innocent children into a living hell. I absolutely know how bad you feel right now.
The constant panic gradually lessened to just bursts of it on a daily basis. I occasionally have the odd moment now. I don't even really know what has changed. I definitely don't ever think about killing my children anymore, thank god. I don't take the medication anymore (but I did find the medication useful and I took propanalol for instant relief of pounding heart etc). I don't have therapy anymore and didn't find it that useful. I never liked the idea of joining any climate action groups because I couldn't cope with being around people who were confirming that I was right. What I really wanted was for it all to turn out to be a misunderstanding and everything would be fine.
What seemed to help was resisting the urge to look up news on an hourly basis. I'd do it in attempt to see that some action was being taken or see something positive, but usually ended up seeing something which made me feel worse. Weirdly, seeing people suffering their own personal problems in life helped me to stop obsessing so much because I realised how many awful, personal crises the people I pass in the street might be currently experiencing while I was spending all my time fretting about things decades into the future. It seemed almost a luxury that other people couldn't afford right now. I also had a cancer scare and it gave me some perspective. I can't guarantee that everything will turn out ok, but worrying so much about it wasn't making it any better and was spoiling the time I did have with them. I realised that I could die in a random accident and all my worrying would have achieved nothing. I also took some comfort from the idea that my reaction was partly fuelled by the fact that I've had an exceptionally comfortable life so far, in historic terms. Throughout all of human history, people have never been completely safe and comfortable. There have always been threats and dangers. I have become accustomed to things always being ok or knowing that an authority figure could help me. Recognising that the feeling of safety and security I have grown up with is actually the exception has helped me to reframe it somewhat. I thought about the fact that I am insulated from many of the worst effects (or will be for some time) while other people are dealing with it now. They probably don't have the time to sit around frozen with fear like I was. I also tried to broaden my horizons and intentionally forced myself to go out and experience new things, which gave me less time for worrying.
I 'do my bit' as much as anyone can in a society built the way ours is. I try not to beat myself up about the fact that I'm not perfect and make improvements where I can.
Apologies if none of that is useful or makes you feel worse, but it seemed that just working my way through my thoughts and finding different ways to frame things helped me come to terms with it to some extent and accept that I can't control it.