To be honest, from one girl to another,my suggestion is to drop him a message along the lines of “I wanted to let you know I am here for you whenever you need me, take all the time you need to relax after exams and give me a call once you feel refreshed, I know you will rock the LSATs, you worked so hard and I am real proud of you. I love you and miss you”
Do not pester him with Hi, how are you messages to nudge him to talk because you feel anxious. Let him miss you a bit and also have him know you are ready to give him his space and have a positive orientation towards things.
Do your bit and go chill with your girlfriend’s, watch a movie, go shop.
He will be back and happier still that he can rely on you to be calm and supportive in these moments. He sounds like a dedicated chap and you are a devoted, sensible girl. It will all be alright soon.