4 weeks of burning pain on right side of head and face. Throbbing gums on right side. Sore neck and shoulder blade all on right side. Nothing relieves?

TMJ syndrome. Your pain may be due to a combination of problems, Your dentist may recommend a variety of treatment options.
Burning pain. Possible causes are viral infection (shingles), fungal infection, nerve irritation ( trigeminal neuralgia). Check with your dentist, oral surgeon and physician for a diagnosis and treatment recommendations.
Pain killers. Pain killers fon't cure problems, only mask the symptoms. Pain may be due to infected teeth and/or gums, infected sinus/throat/ear, trigeminal neuralgia, TMJ Dysfunction, etc. Start with your Dentist to evaluate for oral or peri-oral causes. If normal, the next stop would be your General Physician.