Hi. You are right you need to go to the GP and get your thyroid function tested. That's the first step at least.
I am currently hyperthyroid, having a hemithyroidectomy in the next couple of weeks. I fully expect at some point in the future to go hypo and have been reading up on it. I get the impression that the NHS view on hypo is that it is easy to treat and can be managed by GPs rather than specialist endo's so you may have to push for a referral if you want one.
Other things to think about - if you haven't already - are the way you take and store your thyroxine (T4) tablets. I have an excellent book at home which has really helped me over the last year, The Thyroid Source Book by M. Sara Rosenthal. She is a thyroid cancer survivor and an academic. It is very practical and no nonsense, it deals with the myths and mumbo jumbo and has an excellent section on how to take thyroxine.
One interesting thing she says is that forgetting to regularly take T4 seems to be the most common reason for people to become hypothyroid while taking the same dosages that were previously enough to bring their TSH to a normal level.
She also recommends taking thyroxine first thing in the morning on an empty stomach as food in your stomach causes a slight decrease in the amount of T4 absorbed by the body. Soya, particularly in liquid form should be avoided for at least three hours after taking thyroxine as should hot liquids especially coffee.
I read elsewhere recently that taking thyroxine last thing at night is now being recommended, so that might be worth looking in to.
Other things that can slow absorption of thyroxine are taking iron and calcium supplements, antacids, some heart medications and lithium.
Any heat can quickly inactivate thyroxine tablets so they need to be kept in a cool dark place.
Sorry if you already know all this but I have met hypo people who have never heard this and certainly not from their GP.
You may also have another condition that is masked by the thryoid and you should ask your GP if anything else might be causing some of your symptoms.
As for alternative therapies, my personal view is keep active and take exercise when you can. Treat yourself with some massage, yoga or shiatsu, the sort of thing that needn't cost a lot of money and which has physical and emotional benefits. Maybe look at your diet, a low-GL diet can be a healthy.
Sorry for the long rambling post, particularly if you know it all already. I really hope you start to feel better soon.