I would definitely see a doctor, just to be sure, but it sounds like you have post-concussion syndrome: https://www.mayoclinic.org/diseases-conditions/post-concussion-syndrome/symptoms-causes/syc-20353352
Make sure to tell the doctor about all of the workouts you're doing, to make sure you're not perpetuating the problem by doing a specific exercise. Hope you start to improve!
I agree with the message above. I would definitely go get check out by a doctor, just to be sure that everything is alright. I wouldn't risk playing any sports or doing anything that you could get hurt doing, it could make your head worse because of your injury 5 months ago. Try to get to a doctor before going back to school. It could just be an aftermath of your concussion, but I would also recommend taking an advil when you get headaches until you can get checked out by your doc.