I can't think of a word that describes somebody who is under a lot of pressure. The most satisfactory word I can think of is "strained," but that isn't very rich. It can be an adjective, noun, or even verb, as long as it describes somebody who is under a lot of pressure. I'll tell you if I end up using your suggestion. Thank you.
You could say that they're under the gun.
In a stressful situation that requires quick action, often to meet a deadline; under pressure. https://idioms.thefreedictionary.com/under+the+gun
"Under pressure" (which you've already used in your question) seems to work too.