Hello~It sounds thyroid-related to me. The temps are what makes me say this. If possible, go to the pharmacy and purchase a basal thermometer, take your temp under your arms first thing in the morning without moving around before, leave it for 10 minutes. If the readings are below 97.6, you have a low thyroid. Do this for 5 days in a row. I would then take the info to your doctor and ask for a complete thyroid blood test to be run.
What kind of thermometer are you using? differing methods are more reliable than others.
If you are constantly having numbers under 96 then you need medical help. Getting under 95 can lead to hypothermia. Numbers as low as 93 warrants an er visit. Anything under 93 is a medical emergency. Have you been checked for infection? More serious infections can occur even without symptoms and cause severely low body temp. You don't have to have a fever to have an infection like everyone thinks. Whenever I get an infection my temp always drops, rarely does it rise. It could also be hormone related or Wilsons Syndrome, but thyroid temps rarely drop that low. Wilson's temps on the other hand do drop quite low. Considering you are indeed also having fever symptoms like sweatiness, I would certainly go to the doctor and get you WBC checked at the very least. The times I've had a severe UTI or tooth infection my temps have always dropped and never increased. I know that if my temp drops in the low 97s that I have an illness going on somewhere.