They are right. There is a possibility for a rebound effect by taking OTC too often.
As to the headaches, I get a lot of different kinds.

Ice pick headaches, that feel like someone sticking something sharp into my head. Sinus headaches that can lead to sinus migraines (the feeling of head pressure helps me realize it's yea old allergies). Stress migraines that begin as a band around the head (knock on wood haven't had one of these in a long while).
I'd try the no med week to see if it helps. Hope you feel better soon.