The soft tissue work recommended by your chiro is worth considering. If the therapist finds a spasm that responds to myofacial release, it can be helpful toward healing/strengthening your neck.

But, if head movement causes balance issues, it can also be a vision problem. Find a NORA behavioral optometrist for an assessment. Health Care Locator Custom.

A large part of balance is visual. If you eyes dart as you move your head, you balance will suffer.

Years ago there was a post of neck strengthening exercises on NT. I'll look around as see if I can find it. It helped some.

Unfortunately, there is no such thing as "Get this treatment and you will be better." Finding a therapist who understands and knows how to work with your needs can be a trial and error task. Don't let a therapist talk you into continuing something that does not appear to help. Try another therapist.