I am sorry you are having such difficulties. I would suggest you go to a gynecologist about this. Heating pads always help me when I have bad cramps.
First of all let me say I am deeply sorry for this and I am telling you how sorry I am because I know that people do not really understand what a person with severe menstrual cramps goes through. They see it as "oh...its just your period? well that's a normal part of life" and it is not very nice. I have the same issue as you are outlining and I have had it for years. I am no medical expert so I do not want put any fear in you because each person and their body is different. I have the same kind of symptoms -- headaches, dizziness, vomiting, diarrhea and yes -- pain, pain, pain for quite a few days, the first and second days are the hardest. I am 24 and after years of suffering, I was on the pill as well and pain killers? pssh, they don't do a thing. I finally went to my doctor about it and he sent me to get a pelvic ultra sound and they found a fibroid. They told me that is why I had so much pain and other issues. Now, all you need to do is visit your doctor and get some medical assistance especially since you are going to be living on your own.
There are other pain management methods that I can suggest. Over the counter drugs won't help much but for me I have found Advil Liqui-gels to be more effective than the others. There are also some prescription medication that you can try such as Diclofenac, Zeredol, Panedine, Ibuprofen etc. You also want to take the pills days in advance to counteract the pain because once it hits, its a home-run. Ask your doctor to prescribe something for you. There are also natural pain relief methods that can be useful such as trying a healthier diet, exercising more (even walking), sticking to more hot fluids than cold fluids (that's a challenge for me), taking a hot bath when the pain hits or using a heat pad on the area that hurts.
Keep safe and good luck with school :-)
For me, the only thing that takes three edge off the pain is prescription naproxen (550mg), which is generic for Aleve. I always get a refill from my gyne when I run out. And I use the thermacare heating pads because it helps a little. Nothing takes away the pain 100% but those 2 things help me.
If you go the Advil route, they say it's better to start taking it a couple of days before you think you'll be getting the worst of the pain. Because it's easier to stop the pain from increasing once it's already severe than it is to try to lessen the pain once it's already really bad.