fT3 is what works with your T3 receptors in your cells, so its more useful to have free t3 [fT3].
see the thyroid basics sticky, it should explain
if you don't get very specific, you doc probably will do T3, T4 !avoid!
You are iodine deficient. Which is good in a way...
Iodized salt is rarely used in restaurants and prepared food. So its up to you to use iodized salt in your salt shaker or take iodine supplements. See that sticky. Hopefully you can restore your metabolic rate and temperature with iodine replenishment. You could try that first then do labs and have those measure you in that state.
IT IS VERY IMPORTANT THAT YOU TAKE SELENIUM WHEN DOING THIS.
And read that sticky.