Welcome to Thyroid Board
Checking FT3 would be a good idea - medicating with T4 only just doesn't work for everybody. You might need added (synthetic) T3 or you could do better with natural thyroid hormone.
You could also look into is pH balance of the body. A controversial subject yes, but cutting back on for example coffee, sugar and artificial sweeteners and eating more veggies certainly doesn't hurt anybody. I can certainly relate to the exhaustion and total lack of enthusiasm which I'm still fighting (changing physical activity is also a factor) but with the help of my alkaline drink and improved diet (I don't follow any specific pH balance diet religiously) I certainly feel better than last spring.