There is definitely an interaction between estrogen and thyroid stuff which I think deserves to be explored and trumpeted way more.... There are undoubtedly many other women like us out there who haven't yet figured out the thyroid involvement and are just suffering. When I understand what the heck is going on, I might contact Louise Newson and ask to go on her podcast

just to get the word out more.
For me, this all began only when I started HRT. I'd had the cold feet and slow metabolism, constipation, dry skin for years and years before - but I just thought that was me and got on with life, given my TSH and free T4 was always normal and doctors didn't seem bothered. I was on the desogestrel POP for 10 years which kept all my hormones quite low. (To suppress endo.)
Then I came off the POP and started HRT and after 3 weeks, I got all these terrible nighttime symptoms I'm still battling. And they seemed to get more intense with each dose increase, so I ended up in A&E when I went to 75 for the first time. Of course at first I thought it was the HRT and the estrogen and I spent ages carving up patches and slowly slowly raising things... to no avail. I even stopped it totally for 5 months, thinking I'd get back to how I was before, then - but nope, that wasn't happening either - I just got low estrogen symptoms as well(!). So I went back on it again and this time things didn't get any worse when I increased. I actually seem to absorb it very poorly so need quite a high dose to get a decent amount (now on 5 pumps of gel).
But I faffed around with all that for sooooo long. Then I did B12 shots every other day, buying them online from a German pharmacy and self-injecting at home just to rule out B12 deficiency. I checked folate and D3 (fine). I had an iron infusion in February as I couldn't get ferritin over 62 orally. (Now it's 211 and I still have these symptoms.)
Which is all to say, whilst it began after starting HRT, I now believe it was the HRT (estrogen) and thyroid interaction which caused all this. My thyroid was struggling and only just getting by and adding estrogen in and it couldn't manage....