I'll jump in with CaveDweller's comments. I too, first shaved my head in January. I did not have issues with cold related headaches, but certainly during the initial period of adjustment to shaved or very short hair, the use of a cap would be recommended, even if you don't normally like hats.
I suspect that you will adapt and get used to the change of temperature and the use of a hat will be quite temporary. I used to live in Alaska. I moved there from Michigan. The day I arrived, it was -50F. Sure it was cold. I put on may layers. As I got used to the cold, there were less layers. Then many years later moved to North Carolina. Wow it felt HOT, but I've adjusted to that, too, and now feel very cold at 40F. You get used to temperature changes.