TSH is a pituitary hormone that is affected by so many things that, at best it is only an indicator, to be considered along with more important indicators such as symptoms and also levels of the biologically active thyroid hormones Free T4 and Free T3. You have many typical symptoms of being hypothyroid. Also, your blood test results show that your Free T4 is way too low, even below its range. Your Free T3 is less than 10% of its range, which is also far too low.
The biggest problem you face is getting the doctor to diagnose and treat you. The reason is that the NHS protocol for diagnosing potential hypothyroidism is to test for TSH first. If it is within range, usually the doctor will not test further and will tell you it is not your thyroid. If they test further it is usually only for Free T4 and if in range, they will again say it is not your thyroid. I really don't know what you can expect from your doctor since your TSH is within range and Free T4 slightly below range.
What you should push for is to be prescribed thyroid medication to raise your Free T4 and Free T3 into the upper half of their ranges, and adjusted from there as needed to relieve symptoms. Such clinical treatment is the exception, however, in the UK.
So you are going to have to do some reading and better understand what you need so that you can push the doctor to treat clinically. Another UK member who was having trouble getting diagnosed and treated told us the following: "What I have learned from my experience is that you have to go to the Dr's office and TELL THEM WHAT YOU WANT and to go backed up with knowledge. You have to tell them that you have done your reading and looked into your condition and care about the long-term treatment of your health and thyroid. If you fight for what you want, you will eventually find someone that is happy to go along with your wishes. But we all have to take charge of our own health, right?"
To help with your discussion with the doctor, I highly recommend reading at least the first two pages of the following link, and read more if you want to get into the discussion and scientific evidence supporting everything that is recommended.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Since hypothyroid patients are so often deficient in Vitamin D, B12 and ferritin, I also recommend testing for those and then supplementing as needed to optimize, D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 100.
Please let us know how it goes with your doctor.