Thyroid testing in the UK is awful. So you have to be your own advocate. When you next see your GP or specialist, ask for a full thyroid panel to be taken. (They probably won't, but it's worth asking). Then, when the results are in, get a print out of them and keep it. The NHS will only take a TSH test unless you ask for the full panel. If you do ask you will probably get Free T4 tested. In an ideal world they would test for Free T3 as well, but these days the labs decide not to bother regardless of the GP request, but you might be lucky.
A TSH over 3.5 is treated as hypothyroid in the US. Here they will let you suffer with "come back in 6 months" nonsense until your TSH is 10 or above. By the time it's 10 you feel like death. But anywhere between about 4 and 10 you can ask your GP for a trial of Levothyroxine to see if it helps. A starting dose is 50mcg unless you are older or have heart issues. It really is hard to get proper thyroid treatment here, but it is possible if you know and fight your corner. Once on medication the aim should be a TSH result of 1 or less not just in range. This is why you need to keep all your test results in a folder and as you learn more you can refer back and make decisions.
Dry skin, dry eyes, dry mucus membranes are symptoms. But there are many, many symptoms of hypothyroidism, over 300. Obviously we don't get them all. We have our own personal selection. Many hypo patients are cold all the time. I am the opposite and often too hot. When mixed with menopause the symptoms overlap and can be hard to distinguish.