Sounds really bad, especially at your age when you want to complete your education and make a life for yourself.
Here's a link to read about headaches.
http://www.emedicinehealth.com/cause...article_em.htm

In your case the diagnosis was of little use as it didn't lead to a proper treatment.
You mention concerns with the neck, and how it responds to cold packs.
This is a clue.
On a scale of 1-10, how would you rate your stress level?
I'm guessing at least 8.
So what you may have is tension headaches, caused by permanently tight neck, back and shoulder muscles.
These contract and pull down on the head and scalp, put the cervical spine under stress, and bingo you have chronic headaches.

Have a think about your stress levels. Are you ever relaxed, happy, calm?

See if you can put aside 20-30 minutes a day to lie quietly in a darkened room with no sounds, or some quiet music, and relax the entire body. You can easily find helpful procedures for this.
Also examine how you put yourself under stress throughout the day, and try to moderate it.
The best time for a body to relax and rebuild is while asleep, so you need to make sure you are getting proper, sound sleep.
Once again, review your stress levels at bedtime. Try a hot bath.
Change pillows. If too high, will strain the neck.
I have had very similar headaches for many years of my working life. Pain meds hardly made a dent.
I have found 2 things that work on them.
1. Remedial massage should give immediate, if temporary, relief. It will at least confirm you are on the right path.
2. Magnesium. I now take magnesium oil daily, diluted in water, and have had a dramatic lessening in my headaches, which were caused by tight muscles in the back. You can also get it in powder and tablet forms, but they are are slower to work. You can also rub the oil into the skin.
Have a look also at your diet. Ensure it is really balanced and full of nutrients. Avoid manufactured foods, highly processed foods, sugary drinks, sugar in general, take-away-foods, etc. Make all meals from scratch and from fresh things.
Take a multi vitamin tablet with minerals.
Modern diets can be very poor these days, so you have to take control and don't trust the food manufacturers.
Lastly, the medical profession rarely considers the role of soft tissues, such as muscles, in the causes of pain, yet they can often be the prime, if not only, cause.
I strongly suspect that you are under a lot of strain and your muscles are always tense, leading to the headaches.