firstly welcome but sorry you have to be here! Your TSH is higher and as you have said within some ranges but not others!! Depends on the lab and what they use.
Most "normal" people have around 1 so compared to that yours is higher and can give symptoms of hypo.
A good range of tests to ask for: The free T4 and T3 (please post all ranges people here need something to go by!)
antibodies (will show hashimotos or graves)
vit D, iron, ferritin,
cortisol, DHEA (shows adrenal fatigue) female hormone panel
Your thyroid/adrenals/female hormones can all be a culprit for the symptoms. All can affect each other! So for eg if your adrenals are sluggish it will slow down your whole body including your thyroid, your thyroid can also affect your body as it is the metabolism and so on!
If your doc will not address this for you, you need to look up either by asking here or doing a search, a good thyroid savvy doctor, preferably one who prescribes dessicated thyroid hormone, these guys generally look at the bigger picture.
Or you could choose the naturopath route and go holistic but go to someone who can run these tests and prescribe necessary meds not just herbs!
It is a bit of a roundabout route but getting to the source is important, and treatment can also take its time so you have to be very patient