There's a balance here. We should^^ all be concerned about the fact that these weapons exist and never let ourselves become casual about the risks involved. It's essential that leaders never adopt the attitude that the smaller, battlefield nukes are a viable way to win a war because of the escalation risk.
On the other hand, I still don't think we need to be panicking about nuclear war right now. I read extensively about the topic because I'm interested in it from a disarmament point of view. While I wish the bloody things didn't exist, NATO's nuclear deterrent is substantial. Nobody knows what is going on in Putin's head, but all the available information suggests to me that he wants power. There's no power to be had in a country that doesn't exist.
I also don't think he's crazy. What he's done is certainly beyond evil, but it's not irrational given the assumptions he appears to have made before the war started.
I wouldn't underestimate the urge to survive, even if you're an evil bastard like Putin. I also have trust in the ingenuity and creativity of the people working to avert such a disaster.
There is an awful lot of irresponsible reporting going on about this. I've read numerous articles where the author has included completely untrue information, so you can't assume that because something is reported on a news site that the writer has the knowledge required to comment on the risk level. They're clickbaity articles designed to make money at the expense of good journalism. They're enough to make anyone feel like nuclear war is imminent, but so much of it is a big old load of balls.
No one can promise that this won't happen. The risk is always there as long as nuclear weapons exist. But alarming as it is, we've seen sabre rattling like this before and much worse, and nuclear war didn't break out. Nixon actually ordered war planes loaded with nuclear warheads to fly towards the Soviet Union's borders and turn back at the last minute during the Cold War, which was MUCH more inflammatory than the sabre rattling happening now. If you can, try to remember that you're safe right now and focus on doing anything you are able to do to support the people in Ukraine. And don't waste precious time focusing on something you can do nothing about.