Going back to student days, I spent 45 years in Academia, including 25 as a professor. I've lived the challenges you describe. Some years back I took early retirement and these days among other things work as a coach helping professionals, including academics and students, deal with the sorts of issues you are describing.
There is no simple fix to the problem. The stress and anxiety that you are experiencing are probably mostly habits on your part. That is not meant as criticism; they are habits that you have been taught both explicitly and by example. (This is not psychobabble. There is a lot of recent psychological, behavioral, and neuroscientific research pointing to the importance of habit loops in all of our behaviors, including things like chronic stress and anxiety.) As above, there are no quick fixes. Dealing with stress means working to understand your own habits of behavior and mind and what triggers them. With that understanding in hand, you can then work on changing those habits. Good short term news is that simply seeing stress for what it is can provide some relief.
There are a few clear physiological effects that can be addressed straight away. The two big ones are exercise and sleep. Get in some amount of intense exercise a day, even if it's just 10 or 15 minutes of intervals on an exercise bike. Also take frequent breaks during the day when you get up and move around for a couple of minutes. Then set a sleep schedule and stick to it. BTW, the sleep is not wasted time. There is good research that shows that when tackling conceptual difficult problems and even memorization, those who "sleep on it" accomplish their goals faster and more effectively than those who just push on through. They also just feel better, which also adds to productivity.
If you are posting here, you are likely the kind of person who would appreciate a deeper insight into this stuff. Our experiences and emotions do not happen "to" us. They are constructed "by" us, and reflect the concepts that we hold about the world and ourselves. This rabbit hole goes extremely deep, into everything from evolutionary psychology to the very counterintuitive functioning of memory, to the role of the cerebellum in high order thought, to the neuroscience of Bayesian predictions by the brain. But you don't have to dig all the way into that to get the benefits.
A few people I would track down who discuss just how fundamental this is to we humans include Lisa Feldman Barrett (author of "How Emotions are Made"), Anil Seth (his TED talk "Your Brain Hallucinates Your Conscious Reality" is a delight), Charles Duhigg (author of "The Power of Habit"), and Jud Brewer (whose company among other things offers good even if cheesy app-based support for changing habits including anxiety).
Social context matters. If your group spends a great deal of time bemoaning and reinforcing counterproductive perceptions and patterns of thought it will make them worse. Bitch sessions, among students, faculty or anyone else are likely to do more harm than good. On the other hand it can be extremely difficult (and typically impossible) to really tackle the problem without social connections that support you, explore the issues, and provide accountability. We are social primates and are not wired to deal with this stuff by ourselves.
A key attitudinal change is to place your own welfare at the top of your list of priorities. The tyranny of the urgent will pass if you let it. You are in this for the long haul.
On a very pragmatic front, time management skills are critical and often lacking. I recommend David Allen's "Getting Things Done." I'm not big on most self-help books, and Allen can certainly be a bit "rah rah" for my taste. But the organizational strategy/workflow he develops (Capture, Process, Organize, Plan, Do) is widely recognized for its effectiveness.
Finally I will note that having open and candid conversations with a boss or advisor can be extremely useful. Chances are what you are saying will resonate; in all likelihood the person you report to has had or continues to have a lot of the same kinds of problems. When someone hangs up a sign that reads, "The beatings will continue until morale improves" is probably talking to themselves as much as anyone else. That is also the person who can directly affect the conditions under which you work and help you better understand how you might be misinterpreting expectations.
Do be aware, though, that advice on stress management from someone with "Professor" in front of their name should be viewed with some skepticism. Given that there is an epidemic of burnout among faculty, be careful of the blind leading the blind.
Time spent as a student is an excellent (and usually relatively low stakes) opportunity to develop habits and attitudes that will serve you well throughout your life and career.