Google oil pulling. It is an Indian Folk remedy for removing bacteria from the body. If you are low on vitamin D I would try and spend time in the sun. That is the natural way of increasing you vitamin D. If you have a problem with your teeth such as bleeding gums you might consider that as a problem. Consider taking as much vitamin C as you can tolerate. Very good vitamin for detoxification of the body.
Ouch! From your symptoms (sinus pain and low-grade fever) I would have assumed that you have a moderate sinus/ear infection. Maybe you do, but your doctors are just too hesitant to prescribe antibiotics. I had similar symptoms a few weeks ago (1 degree fever, sinus headaches, always feeling warm, tired, previous sore throat, only slight sinus drip) and they prescribed antibiotics and it went away eventually. If you can't get antibiotics, you could try Mucinex-D (over-the-counter) to help things drain and relieve the pressure (this worked for me when antibiotics couldn't quite relieve my sinus infection last year).
Another possibility could be TMJ (Temporomandibular Joint Syndrome) where you clinch your jaw muscles and grind your teeth, which could cause face, jaw and neck pain and wear away your teeth.
Or it could be mononucleosis, which gave me a sore throat, sinus/ear infection, fatigue, slight fever, headaches, blocked ears. The quick 10 minute test does not always catch it and they may need to do a blood test that takes several days.
It could partially be stiff shoulders/neck, which could cause headaches (pain is in similar places to sinus infections) and constrict the tubes of the inner ear, blood vessels, etc in the neck (try massaging your neck/shoulders, face, jaw and stretching, and excedrin or advil works well).
These last two don't explain the warmness or cold hands/feet. You could have your thyroid checked out (have them check TSH, T3, T4 and thyroid peroxidase antibody, not just TSH) or maybe other hormones (although these usually have more symptoms). Maybe you have bad allergies (pollen, mold, animals, etc) or the warm weather is bothering you, too. Maybe it could be fibromyalgia or lupus, but these have more symptoms.
Hope this helps.