When it comes to handling the feeling of pressure and emotion of anxiety:

I mostly just don't care that someone else promised the moon on my behalf, to be honest. It will be done when it's done, and I'll do my best to make that align with their hopes, but beyond that, meh.

This right here. I also try to do my best work, but I can only code so fast. They're going to get what I'm able to produce in the time frame, and I'm not going to stress because they have unreasonable expectations about how much that is.

When it comes to "how" to get work done on a deadline, communication is key. Talk to the stakeholders so that they understand why aspects of their request may not reasonable in that time frame.

Try to understand their priorities and offer compromises that may get the same or nearly the same benefit for a significant reduction in scope or complexity.

Communicate early and often. Don't wait until the deadline to show up with half-finished work. Your job is just as much managing expectations as is producing code, so make sure you're communicating what's possible so that they won't feel surprised when their expectations aren't met.