First of all, great news about your scalp:) It's been said that all skin problems are because of Iodine deficiency...
If you search the term "sore throat" on this forum you will see that similar symptoms have been reported by many. My suggestion to you is to proceed at your comfort level.
With your family history I would venture to guess that you do have thyroid problems. I would suggest doing the Broda Barnes basal temperature test...
Besides the hair loss, what is your health history? History of infections, Antibiotics , etc? What is your diet like? History of gastrointestinal distress? Constipation? Dental health? Mercury fillings?