I don't know if this will help, I have a little insight on iodine deficiency, as you probably have read up on iodine deficiency can lead to hypothyriodism and can go undiagnosed due to on a thyroid function test if in normal range goes over looked by some PCP's.
If left untreated, leads to symptoms you described, although numbness I've not heard of more intolerable to cold conditions, cold feet and hands caused by lack of thyroid hormones being circulated. If iodine deficiency is the cause you should try to figure out what brought this on, generally most causes are, diet, lack of iodized salt or multi Vitamin not actually having 150 mcg in it,as You may of read certain foods inhibit absorption of iodine, peanuts, soy, kale, cauliflower, cabbage .Selenium helps on absorption of iodine and if her Vitamin A levels are low this makes absorption issues also, certain anti depressants can deplete iodine levels as well as some type II. diabetes medicine, You may of already read up on some of this?
So things to do to check if she's iodine deficient is check her Iodine blood levels and ferritn as both are also overlooked by some doctors,when checking for hypothyriodism also check waking temperature, then 4 other times through out the day, if low this could very well be an indicator of iodine deficiency and possibly adrenal insufficiency , if you can find Iodine at a pharmacy place a dab on area of skin such as the inner thigh or under arm area as the skin is thinner and more likely to absorb and generally has less hair follicles, if absorbed rapidly 1-2 minutes can indicate iodine deficient.
Does she eat seafood or use iodized salt?
Something to keep in mind try Not to over treat with iodine supplementation as if its Not excreted,can be very toxic and lead to certain types of cancer.
If after you have checked blood levels and body temperature and indicate low, try to avoid named foods that inhibit absorption and eat iodine enriched foods as well as lightly pinch iodized salt in a glass of water.
If temps raise but not quite up to the 98.6 range address adrenals. If iodine deficiency has been undetected may cause stress on the adrenals. Now two problems! Not good as this leads to other hormonal imbalances that may take time to correct
I'm Not familiar with TOS, but it sounds very painful.
I wish You and Your wife the best, I would recommend asking Your doctor to do all thyroid function tests including a rT3 (reverse T3).
Hope they find the cause of TOS
My wife has had an ongoing battle with numbness in hands, feet, and face, extreme dizziness when standing, as well as fatigue, and weight gain. Did a battery of tests with Neurologist, and it was diagnosed as Thoracic Outlet Syndrome. It's gotten better on it's own, well a little anyway, stress can really crank it up we're told, and she works a very stressful job.
We read up on iodine deficiency, and it sounds EXACTLY like what she has!
Any ladies here ever battled this, any insight into the supplements / diet to add more iodine?
Thanks