It's tricky as t4 does vary a lot during the day, t3 varies even more.
I agree with slutty. I find health unlocked can be misleading regarding the t4 and t3 as they don't take into account natural variations during the day. Which is why drs do look at tsh; where drs went wrong in the past is a) recognising that tsh needs to be a tad lower if on thyroxine and b) listening to the patient's symptoms.
There are some people who really do not respond to thyroxine and do need t3, but there's ways of working that out with an endo.
I did find their advice on ferritin was correct for me, something a relative who researches thyroid and iodine confirmed to me recently. I have to keep that above 70.
I do feel wiped easily if a little over medicated, no palpitations, I tend to wake around 2-3 am for an hour or two then feel exhausted all day and need naps. I did feel as cold actually, weeing a lot in the night. It can affect your blood sugar/ insulin levels. I think I got more dehydrated more easily which made me feel crap (this makes sense as when hypo you tend to retain water).
I've recently started to learn that my best level is around tsh 0.8 -1.
I'm taking 175 as I'm pregnant and my tsh was 0.2 though t4 was mid range. Despite them all saying it was fine, I wasn't sleeping great and having bursts of energy then being wiped and foggy headed for a couple of days. I've knocked it down really slightly to alternate days of 150 and 175 and started feeling better in about 10 days. Joints feel better too.
An endocrinologist told me a couple of years ago some people do need to be really suppressed to feel well; I tried it for a long time but looking back I think I was struggling. I was living in a slight chronic fatigue state and I did have muscle aches that matched fibro symptoms.
I've found some advice on health unlocked to be a bit erroneous. Through my own research I learnt that a slightly hyper state means the half life of the t4 is less, which make sense as everything is going 'faster'. To me I feel that reflects the ups and downs I get if a bit over medicated. So I do think there's a sweet spot for individuals. It's trying to work it out though.
But also, you can't blame everything on thyroid. There are so many other things that cause these symptoms.
Only do the dairy and gluten free thing if you've identified a clear stomach issue with those things, or your fibro. I did develop ibs for a while; going dairy free (lactose free really) helped and now I'm fine again. My iron went up during that time, I think as calcium can block absorption. (I was drinking a lot in decaf and red bush tea all day).
Health unlocked talk a lot about gluten but the actual proven link is if you're an undiagnosed coeliac. Coeliacs are more likely to develop an under active thyroid esp if not treated.
And it sounds like your thyroid has totally given up so nothing is going to get it going again. And you are absorbing the thyroxine. Tbh most gut research points to good gut bacteria affecting most things these days including weight, health, even anxiety and mood. So eating a really wide range of foods and some fermented ones really helps.
As said above, get another test and think about what you might do. Ironically I remember in the past my t3 being no different on different doses of thyroxine while the tsh was and the t4 was too.