It does sound you're getting so many carbs (from fruit and milk) that the headache shouldn't be caused by too few carbs. I started low carb last week too, and have had some head aches and light-headedness. I think I finally realized I wasn't getting enough salt to handle my low blood pressure/NMH. I started salting my cottage cheese and meats like mad today, and did a lot better. Plenty of water too of course!
You probably won't get the benefits of a low carb diet when eating normal fruit and milk (and startchy veggies like potatoes), and if you aren't spreading your carbs out throughout the day, you might be yoyo-ing back and forth between eating enough carbs to use that for energy, and not having enough carbs to give you any energy. It takes a couple days for the body to really adapt, so the back and forth could be causing the headaches.
I've been doing about 8 50-100 calorie snacks per day plus a "normal" dinner ... same as what my fiance eats, but minus the grains. It ends up looking like this usually:
7am: a few bites of liverworst
8:30am 1/4 cup of yogurt or cottage cheese,
10am: a little meat, usually leftovers
11am: 6 berries
noon: a good sized salad with some cheese and fatty salad dressing
2:30pm: 15 peanuts mixed with 4 almonds and 6 hazelnuts
4:30pm: liverworst
6:30pm: normal dinner amount of meat and veggies, maybe with a cream-based sauce
8:30pm: 1/4 cup heavy cream with 6 berries (and a little vanilla and stevia usually)
This way I'm getting 2-4 carbs every couple hours, instead of all at once like I would if I ate a banana. I don't get any cravings, and can even resist oreo cookies easily now! Weight has gone done consistently even after the first couple days of water loss.