I had pretty much the same symptoms as you... horrible fatigue, anxiety, brain fog etc. (plus horrible aches and more).
I've been diagnosed with Hashimotos and the thyroxine wasn't alleviating symptoms (no matter what dosage I was taking). I suffered this for a year (and three GPs). The symptoms I was experiencing were crippling (to the point I was wasn't able to work). I'm almost the same age as you. Finally, I was tested for my female hormones (FSH test) too. The first FSH test said that everything was ok (the doctor had no idea...) because it was taken at the wrong time of my cycle (take it day 21 of your cycle to get proper results). Finally, I found out that my symptoms were a combination of my Hashimotos and peri-menopausal symptoms i.e. complicated overlapping symptoms and conditions that effect each other.
Then I when through a whole trial and error process of finding the right hormone mix that suited me. I had immediate relief from the fatigue and brain fog as soon as I started taking estrogen (via patches) but because of the Hashimoto's I had huge problems finding the right progesterone part of the mix (I had terrible side effects - major acne, horrendous water retention/bloating plus more) and found out that natural progesterone (Utrogesten) has bad inflammatory effects for those that have Hashimotos and are better off (strangely) taking a progestin (synthetic progesterone) and my problems were solved with using Femseven Sequi patches. After that, the thyroxine actually seems to do its job and I am feeling soo much better!
I suggest you "bully" your GP into getting not just your thyroid blood tests (TSH, T3 and T4), but you also get tested for vitamin D3, B vitamins, Iron and FSH (female hormones). That way you can know for certain what you need to supplement with. Hashimoto's people are almost always low on vit D3. (I also, got a cortisol test, just so I could eliminate adrenal problems from the equation). Cutting out gluten in my diet did help me and healing my gut (i.e. my immune system) through supplementing with good probotics and digestive enzymes supplements also helped. Also, don't take iodine, (a lot of people get that wrong and complicate matters through taking it).
The other thing that has helped me and what doctors fail to tell you (and when you have balanced the major problem) and you're recovering - is that if your thyroid isn't working properly you also suffer low serotonin and dopamine, which, in a nutshell will make you feel depressed, anxious and unfocused (brain fog). I take these supplements: l-tyrosine (500mg) in the morning and 5htp (50mg) at night - that's a 10:1 ratio (that's really important you take both as otherwise you get depleted in which ever one you're not taking). The l-tyrosine really helps alertness and brain fog and the 5htp alleviates the anxiety (with the good side effect of helping you sleep, if you take to much you'll get very vivid dreams lol). Also, a good B Vitamin complex in the morning really helps me with concentration and focus.
This is just what worked for me. I hope that my experience offers up some options for you, even though it may not be the solution that's necessarily right for you, it may help you undercover (and eliminate) what may be at the root of your suffering.
I really do hope you find a solution that helps you, as I know how crippling these kinds of symptoms can be.
Wishing you all the best and I really do hope you start feeling better.