Chances are you don't have CFS at all, but a thyroid problem.
Yes, do not wait for three months to be re-tested, try six weeks, and next time you are tested, ask for a full thryoid panel - TSH, FT4, FT3, TPO and TgAB. This is because TSH is not a thyroid hormone at all, it's a pituitary hormone and you cannot tell from a TSH result alone what is happening with the thyroid. You cannot tell how much pro-hormone is available, whether you are able to convert it to enough active hormone or whether you are under auto immune attack.
While you're at it, ask for a full iron panel, ferritin, folate, vitamin B12 and vitamin D. The chances are that some of your vits and mins are deficient and fixing those may make you feel a deal better - it did with me.
All these tests are available on the NHS. You might have to bang the table and your doc might have to ring the lab and bang the table himself, but they are all available and they are proper diagnostic tools that it is reasonable to ask for.
I could recommend something you can buy over the counter that would support your thyroid, but that would reduce your TSH and possibly delay a diagnosis, which I guess is not what you want.
Finally, get a copy of your test results, with the reference ranges - every time. That's your data and you are entitled to it. If you do that and keep a spreadsheet, you will be able to pick up patterns and changes much faster than a GP flicking from screen to screen.