Symptoms such as the what you are having should not be ignored. I think you should see a doctor very soon. The best type of doctor for you to see about this problem is a primary care physician
such as a family doctor
or internal medicine doctor
Our body temperatures have a tendency to drop somewhere in the middle of the night by about 1 degree F. Despite this, I don't think that being freezing cold to the point you are shaking is normal. Chills like the ones that you are experiencing are sometimes associated with infection. However, since they have been going on some time, there are not that many infections that are possible. The night sweats are also bothersome. While there are many possibilities, you must be checked out for a type of cancer called a lymphoma which can cause fevers chills and night sweats.
I suggest that you schedule an appointment with your primary care physician. He or she can take more detailed history of your night time chills and sweats, and perform a thorough physical exam. I also suggest that you keep a log of your symptoms from now until you see your doctor. Depending on the results of your physical exam, you doctor may want to do some blood tests or even some imaging of your body.