I had/have this same addiction. I believe it's the sugar / caffeine combo. If I drink just one Dr Pepper or Coke, I don't get a headache - but that's only around 40 to 50mg!. If I miss that can, by noon I have a headache.
But even though I think the sugar and caffeine work together somehow, I have found the solution to withdrawal is caffiene pills. A 100 mg caffeine pill in the morning and no headaches. For some people, maybe you need 200mg. Do that for a week and then start cutting your pills into halves or quarters and work your way down to 75mg, 50 mg, 25 mg. At 25mg for a week or two, you're ready to go cold turkey, no migraines.
You would think a similar thing would be possible with soda cans - get yourself down to one can and you should be ready to quit. But that's, first of all, incredibly challenging. One can leads to two cans, and maybe I'll just have another with dinner since I already had two today, and so on. And second, it doesn't seem to work - at least for me.