As well as getting thyroid function tests, it would be a good idea to get some other things tested too as they could be contributing to how you are feeling. Good iron and ferritin levels are essential for your body to be able to use thyroxine, regardless of whether it is liquid or tablet form. Your iron should be comfortably within the normal range and many people feel best with a ferritin level around 70-90. In fact there was some research last year that found that women with a ferritin level below 50 benefited from iron supplements. Dealing with a potential iron problem may help reduce your symptoms as your body will be able to use the thyroxine better. You may have too much thyroxine in your system purely because your body isn't using it well. Thyroid function tests and iron and ferritin tests should help to determine if this is part of the problem.
People with hypothyroidism in particular need really good levels of folate and vitamin B12. The bottom of the healthy blood test range is around 4 which is much too low. Many people feel better with it above 12, particularly if they are supplementing with folic acid or folate. The bottom of the healthy blood test range for B12 if often below 200. Few people feel properly well with a level this low and other countries recommend a minimum of 400-500 depending on the country. Low folate or B12 can leave you feeling really rotten and symptoms are often confused with hypothyroidism. It is worth getting this checked out as well.
Vitamin D is another important one. It is actually a pro-hormone involved in over 300 metabolic processes. My doctor says that all women should be supplementing with vitamin D between October and April regardless of their health. Deficiency is very common and has symptoms in common with hypothyroidism. Some of the most common symptoms are fatigue, depression, chronic joint/muscle pain, allergies, weight gain and so on. This is most definitely worth getting tested. My chronic pain was partly due to a vitamin D deficiency, as was my depression and fatigue. Again these symptoms are common to hypothyroidism so vitamin D doesn't even get thought of by many doctors.
The last one to look at is cortisol. If your cortisol level is too low or too high, you won't be able to use thyroxine properly and you will feel how you describe above. The only test available on the NHS is the random cortisol blood test. If you have this test first thing in the morning, cortisol should be toward the top of the range. If you get the test in the evening, it should be lower in the range. To get a better idea, a 24 hour saliva cortisol test is best but those you have to get privately so concentrate on the others first if you don't want to go down that route yet.
I know this is a lot to take in but it is worth getting these tests if your doctor will do them. If he is saying you are on enough thyroxine and that this isn't the cause of your symptoms, he should be checking the above things at the very least.
I hope you get somewhere with this. You deserve to feel well.