I've had shooting pains and frequent headaches over many years after having several whip lash injuries in car accidents and was eventually referred to a neurologist. Mine were more chronic than severe; sound some different from yours. Initially, I had the shooting pain but this subsided to chronic dull headaches and stiff neck. Diagnosis was muscle tension and not a pinched nerve, etc. I had physical therapy which didn't help much. Chiropractic helped some but the headaches always returned. I've improved my posture and have fewer headaches.
I've had x-rays and CT scans but only after the headaches became chronic.
Sleeping in a cramped position, turning my head while keeping my body straight, lifting heavy objects all cause headaches. Sometimes I still have the sharp shooting pain at the beginning. When the headache started with a sharp pain it took several days to a couple of weeks for the headache to subside. I've been told to keep moving, use heat and ice as well as ibuprofen for pain. Trying to not move your head will cause muscle tension. Over the years I've learned that a prescription muscle relaxant will speed the pain away but I have to be consistent in taking it.
If you've had this headache and shooting pain for just a week, I suggest that you've tried too many things without spending enough time to allow them to work. It's unlikely that you've had a major injury since you don't recall an event that would result in injury.
I agree that if the chiropractor did not take an x-ray, I'd ask that he take one before continuing treatment. However, if he can discuss sensible reasons for not needing an x-ray then I'd accept his judgment.
I suggest that you call the neurologist's office and ask for advice. I'm thinking that they have an advice nurse just as generalists do.
I know how debilitating headaches are and how anxious you are to be rid of it. It's likely that you have strained a muscle and will just have to wait it out.