I am very sensitive to cold, drafts, etc...

Cold: Lack details but many problems causing cold feeling. First people think of is thyroid- easy check, but these pts cold to core. Rectal tempurature can give clue as it reads core temp. Other causes can be sympathetic nervous system which automatically regulates heart beat, breathing, etc and opening and closing blood vessels, sweating.Etc. Medications like beta blockers. Consult physician.

Sensitive to Cold: If this is a "new" feature for you, it could represent an underlying condition: http://www.Nlm.Nih.Gov/medlineplus/ency/article/003095.Htm. As you note, some thyroid conditions and anemias may be at play. Check with your doctor to rule these out first.