Just skim read this but yes tsh must be much lower than that, I know now I feel much better below 2,but it does take a while to stabilise and for me to feel like my body has caught up with the levels - sometimes as much as two or 3 months for the muscles to recover and your cognition too, so you may still be on catch up. When my tsh was 7 a few years ago and I was working flat out I actually started slurring my speech and swear I shouldn't have been driving. You could try a couple of different gps first as I am always astounded by the difference in their knowledge and approach. You actually have a right to be referred to an endocrinologist if you can face the battle with the gp, for a second opinion. Two obstetric consultants recently told me I could have seen them sooner if id requested.
I noticed improvements in energy with vitamin b complexes eg berocca or other - cheaper version in Asda but they add calcium so don't take within 4 hours of thyroxine as it interferes, I think iron does too, but no iron in berrocca.
I wish I'd joined the British thyroid foundation sooner, a wealth of support, and if you can get to a meeting it can be so helpful to hear others stories. A local.consultant at my local meeting hands out a chart for you to record results and muscle state, plus mood etc for him to collect back in (as he knows the lower end of tsh is better!)
The other thing is iodine. My cousin is a scientist and researches this in Edinburgh and gave me lots of advice recently (as did guest obsetric consultant at recent btf meeting on pregnancy as iodine nec in preg). Dairy is the best source, esp fat free milk and yoghurt, as well as mackrel. 2 to 3 dairy servings a day (or more if you fancy it - I love dairy!) and a couple of oily fish servings a week should be fine. sugar free natural yoghurt is best and I noticed my mild reoccurrant thrush disappeared when I started including some daily. Never take iodine/seaweed supplements; my cousin is currently researching them at mo and said its hard to get ones that aren't polluted with mercury etc and not least too much iodine is as bad as none and the levels are just too much in holistic supplements - it can cause lots more problems.
When you do finally feel better (you'll find impossible until you do) try increasing your exercise, I've found improving my aerobic capacity and also a bit of strength (ashtanga yoga, swimming, even light weights) has helped when I've then wobbled a bit - I really notice the difference in my muscles now when hypo but ive recovered my energy more quickly as a result of being fitter before hand (not my brain though! - no idea what helps that other than rest until it gets better!)
I notice my levels changing most in my (sorry tmi) toilet habits, and now go straight to the doc when I start to get very constipated. It's worth looking into different fibre types, some just make you more compacted, (and painful wind) for example brown rice seems to be bad for me when hypo but porridge is good - veg seems to be the best, a recent discovery being beetroot!
Hope you feel better soon, people with good thyroid function never have a clue how it makes you feel xx