Some people above have mentioned already, but a great way to study is to do practice exams (if you don't have any, create your own questions in an exam-format and such). It seems like you're struggling more with the psychological aspect of test-taking rather than the actual memorizing of material.
Try sitting down in a room that is similar to your exam testing rooms and immerse yourself in that environment. Give yourself a time limit, start at an arbitrary time (eg. if you're studying in the afternoon, start exactly at 2pm and finish in 2 hours or however long the exams typically are). Don't look at answers if you get stuck, just treat it like an exam. Some people prefer to finish written answers and transition back to multiple choice based on time, so try that too if it helps. Ultimately you're trying to habituate yourself to testing, which can be done with some practice. When you write your actual exam, imagine that you're just doing another practice test. Just remember that exams are, ideally, a reflection of what you already know. You've done the practice problems, you know the material, it's just a matter of putting it on paper. Best of luck!