Thank you for your pictures. When we see segmentation of the styloids, as in your case, it’s more that the stylohyoid ligaments have calcified which creates the appearance of styloid elongation. Your left side obviously has a significant & very curved section of calcified stylohyoid ligament that extends below your mandible. The thick “joint” in the right one plus it’s length also make it look quite alarming. It’s no wonder you’re having the awful symptoms you have. I’ve annotated your first image to show the odd curve you have in your cervical spine. Typically we either see a proper lordotic curve, military neck (no curve) or a reduced lordotic curve. Your neck is straight in the top 2/3rds & has retained a curve in the bottom 1/3rd. That “interesting” cervical alignment could also be contributing to your symptoms. The lordotic curve can be restored but it takes time & patience. We recommend waiting to work on that until after the styloids have been dealt with but you could begin thinking about that now as a future project. Here’s a link with some excellent information in that regard.
You should request a CT scan w/ contrast & done dynamically if you can get that when you see the ENT you’ve been referred to. Any ES surgeon you consult will want CT imaging. Hopefully the ENT will be willing to at least refer you for that. 
I’m really glad that the Naproxen you’re taking is by Rx from a doctor & that you’re alternating heat & ice to help manage your symptoms to some degree.
