I had an MRI that showed nothing unusual. I've never seen a chiropractor.
If someone found something out of place and put it back, and solved my issue, I would be the happiest person on Earth for a while.
My neck also felt unstable. Like my head would "slide off" if I tilted it wrong. My cervical vertebrae also would click in certain positions.
It was actually a physical therapist that fixed it. I had gone to them to "stabilize" my neck. They noticed that I couldn't turn my head to the left as far as I could to my right. We did ~2 months of exercises and one day they tried manipulating one of my cervical vertebrae and it was like night and day. I felt a rush of blood up into the left side of my brain and that night I slept great. It took a couple more weeks of exercises to stabilize the new position (other things started to move now that it was better aligned). And I haven't had any issues since.
I hope it's the same for you! I understand what it's like and it's awful. I think in my case it was more of a rotation problem so that might not show up as well on x-ray. Idk what other imaging would be better (if any). It might be worth getting a physical therapist to check out your neck mobility and see if there's anything noticeable. Best of luck.
Also you might need to try more than one physical therapist. Some are much better than others.