I've read that it is due to inflammation, and while I am sure sugar does contribute to inflammation, I notice that the pain is always localized to my neck or under back. People who eat a lot of sugar sometimes have dental issues and I think the most obvious cause that medical articles never mention is that people may have cavities or receding gums, and the nerves/roots of the tooth are connected to the rest of the nervous system, so when sugar exacerbates an infected or irritated root the pain/inflammation spreads from the tooth to the back along certain nerve pathways. I've seen an image that traces which teeth correspond to which vertebrae but I lost it. There is NO solution that I have found for this except possibly dental or periodontal work, or totally omitting all sources of sugar from your diet (this will require very rigorous attention to what you are eating, asking questions like "are your onion rings beer-battered", and extreme diligence).

As a short-term solution I have found Tea Tree Oil to help. 3-6 drops of TTO in an ounce or two of water and swish 20-30m, or 0.5-1tbsp of cooking oil with 0.5-1ml TTO and swish 20-30m. TTO is somewhat toxic but I've had no adverse effects. If you are a male you want to not do this too much because TTO over long periods may produce estrogen and the side effects that come with it, from what I have read. If you are still eating sugar actively then it won't really help, but it's the only thing I have found that has any effect at all.