I do react badly when I'm very ill and thyroid labs drop considerably and take a few weeks to go back up. Feels like I need extra to cope with the illness but this is generally alongside a high temp/fever.
As mentioned, if you post your blood test reports, people can advise better whether there's things that need working on or even see if they suggest you are not responding to T4 etc etc. Not much we can suggest without these. I don't respond to T4 as can't convert so am on T3 instead so there are options if this is the case although hard to come by. Also, how long have you been diagnosed and what dose are you on
Would be worth getting your vitamin levels looked if not already - Vit D, B12, iron, Folate and post this as well.
I know how it feels, I was diagnosed with fibro and hypermobility and siatica and restless legs and water retention and had a lot of inflammation in my joints and so on etc etc. To be honest, everything used to hurt - actually my nose has never hurt much lol. All went away when went Gluten Free. May not work for everyone but quite a lot of hypothyroid people find it helps so if you're really desperate to try anything, might be worth giving it a go
Vit D deficiency with levels under 30 causes a lot of pain in my bones also - feels like they're bruised and very sore to touch.