The anxiety is not doing either because it can't. You are over-analyzing your body because anxiety is making you fear something is wrong, so you keep track of every little pain you experience now whereas without anxiety you would just ignore an ouch here and there - we all get them and ignore them.
You may actually have a physical issue like a sore throat from an infection or dryness, but you would have to see doc for that diagnosis, but anxiety is not causing it.
Neck soreness is a very common anxiety symptom, but not the throat problem. When you say always, for how long are you talking about? If you have had a sore throat for a while, you should see your doctor. As for natural remedies, it depends on for what. You need a diagnosis to treat with natural medicine just as you do with allopathic medicine.
I have panic attacks/GAD. Always with my panic attacks I had throats issues like it was closing or lump in my throats feeling. Classic signs of panic. I finally went to a therapist and found a book off internet that helped me understand panic so much. Teaches you how to deep breathe to calm down. If you learn more, it helps you calm down. The book is "panic attack workbook" by Dr. David Carbonell. I think I've had every symptom there is out there and I'm doing much better with knowledge. A therapist really helps too.