You've put your finger on the problem. Anxiety. Help yourself by addressing your anxiety. A few things I have done: told the instructor I have terrible test anxiety. I received reassurance and whatever pointers the instructor thought it was fair to give me. For my really, really bad math anxiety, I read up on math anxiety and found lots of great reading about it on the internet. That helped a lot.

I found that the commotion in the classroom right before the exam put my anxiety through the roof. So I timed my arrival for right before the exam, where possible. Where it was not possible I told the instructor about the effect the commotion had on me and how I was going to tune it out right before the exam. That instructor allowed people to be rowdy for a bit, and then said specifically to calm down because 'here were the instructions.' Then in a calm quiet voice she gave them. I tuned back in long enough to listen to this. This helped me.

For essays, I always plan ahead a few major points I know I want to discuss over the topics I plan to write about. You're thinking "I don't know what the topics will be!" Yes you do. They're the very stuff your class has been studying the whole semester (this week, whatever applies to the time period). If the instructor has emphasized, for example: 'the problem began after the treaty was signed' I'd mentally squirrel away a few points about what was significant about the treaty, what the problems were and what they led to.

Where there are gaps in my memory I have written "the details have escaped me" or something along those lines and written what I do know. Better to get some points than none.

Accept being imperfect. You have your own unique point of view. You know what you know. And you know more than you realize. Let it flow out of you.