Did you have this pain before surgery as well as after? And if before, for how long?
Can you parse out where the neck pain is? What level and on both sides?
The spinal accessory nerve goes to the traps. The traps attach to the bottom of the skull and the greater occipital nerve comes through the top of the upper trap. Could it be that both the spinal accessory nerve and the (or one of the) occipital nerves is causing your symptoms?
Do you have a local Physiatrist (Physical Medicine and Rehabilitation doctor) who can assist?
I would look into botox of the traps to get them to settle while you are healing. And maybe more immediately try Baclofen or Flexeril to get some relief, and check in about Gabapentin (try it again) for nerve pain (thinking occipital nerve pain (?)).
Occipital headaches explained – Dr Ivan Ramos-Galvez.
“Occipital neuralgic pain from irritation of the Third Occipital Nerve is a common presentation of referred pain from wear-and-tear of the spine at the top of the neck. The symptoms include neck pain with painful movements of the neck, which not only are lessened and accompanied by crunching noises, but also refer pain into the back of the neck and sometimes as far forward as the eyes. Muscle spasm of the Trapezius Muscle is often seen. It is a natural reaction to the wear and tear of the bone frame of the spine. When this happens, the small nerves that will feed into the Occipital nerve will have some pressure around them that will translate into pain. These symptoms often affect both sides of the neck/head.”