You know, I'd never heard of this before until a few months ago, and now you're the 3rd person that I've heard that has this.
The most likely reason for these symptoms is drying out your throat, due to breathing through your mouth instead of your nose. Especially if you run in dry, cold weather, breathing through your mouth dries out your throat lining. Other environmental things that could cause this are dust or pollen in the air. If you're allergic to these things, it can trigger swelling in your vocal cords, called vocal cord dysfunction. The medical conditions that could be exercise-induced are sinusitis, gastric reflux, and asthma. All of these could lead to a cough and shortness of breath with exercise, but sore throat alone would be unusual.
I'd suggest breathing through your nose as much as possible when running, and wearing a scarf or cover over your mouth and neck. If that doesn't work, try an antihistamine before you go running. If you develop other symptoms, you should see an ENT doctor to look into your throat with a camera.