First off, are the T4 and T3 tests Free T4 and Free T3 or are they Total T4 and Total T3? They aren't the same tests and don't provide the same information.
Secondly, do you know which antibody test you had? There are actually 2 of them to test for Hashimoto's, so if you didn't have them both, you could be misdiagnosed.
That said, both your T4 and T3 are too low in the ranges. It's typically, recommended that Free T4 be maintained at about mid range and yours is only at 24% of its range. Most of us feel best with Free T3 in the upper half to upper third of its range and yours is only at 28% of the range.
You should make sure you get both of the antibody tests in order to rule out/confirm Hashimoto's.
It would also be helpful if you could tell us some of your symptoms. There are, typically, about 26 that are routinely attributed to hypothyroidism, though some people have others as well.
Many people with hypothyroidism also have some vitamin/mineral deficiencies, which can cause symptoms of, both, hypothyroidism and M.E., I'd suggest that you get tested. The main ones recommended would be vitamin B-12, vitamin D and ferritin.
If your doctor isn't convinced you have M.E. and since your thyroid hormone levels are so low in the ranges, you could try asking for a dose of replacement thyroid hormones to see if it helps you feel better...