You may have a problem with caffeine, if you only drink it in the morning. If you space your intake throughout the day, you should be able to rid yourself of headache from caffeine withdrawal.
Food triggers can be tricky, especially with prepared foods, i.e. foods that have multiple ingredients. For instance, a well-washed apple is a clean food, but apple sauce may have other ingredients in it. The best and most successful way to get a good hold on food triggers is the "put back" method. Basically, you don't consume anything but water, and then slowly add a single food or liquid back into your diet every few days. It's very drastic, but can be fairly accurate in highlighting what foods, liquids, spices, or condiments that may be causing you problems.
Personally, I don't have food triggers. My major triggers are environmental, and since I can't control the weather, I'm screwed a lot of the time. I hope you can find your triggers and get the correct medication that can prevent or at least abort your headaches successfully.