Hello and thanks for trusting me to help you. I am a board certified consultant with over 30 years of experience.
I understand your concern.
Well as you describe,possible causes of such pain may include Cervical spondylosis/stenosis-nerve compression in it's path due to osteophytes formation from degenerative arthritic changes,bulged disc in neck.
You should see an Orthopedist or Neurologist for a clinical examination and undergo X-ray/MRI of cervical spine and shoulder joint.
As you had already tried anti-inflammatory medications with little relief so your doctor may prescribe opioid pain killers,steroid injections for pain management.
Physical therapy can be tried.
Warm compresses with a heating pad on affected area and back of neck.
And make an appointment with doctor.
Let me know if you have any further query so we can discuss it thoroughly in every aspect.
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It’s my pleasure to continue this conversation until you satisfy.
Waiting for your response.