I know what you mean. I only knew about the parathyroid test because when I lived in Cardiff I had a really good gp there who was up on this. I then started reading up and doing my own research and came across this site which is brilliant as I have learned so much from here.
If you have not had a thyroid antibodies test done, could request one from your Gp. I had my first one ever just before Christmas and after being on thyroxine for more than 5 years and on a high dose, mine are high.
The following tests would be helpful to you:
Also, B12, ferritin, magnesium, zinc (if your gp will do it, mine said no as its only a trace mineral - I may not agree with that though)
If you're symptoms have worsened since decrease in dose, sounds like you need to increase back up to a level that makes you feel better.
Gp's should be doing TSH, T4 and T3, also antibodies. Most, including mine only test and monitor on the TSH, which is the worst way to treat hypo. That lead me to taking things into my own hands x