Hi, general crappyness is definitely as good a word as any for this I think! Just so you know .25mcg is pretty much a nothing dose, .50mcg is a starter dose. The TSH of 3.861 does suggest and match your symptoms as a normal person would have a TSH of 1, the higher the TSH the more crappy you feel in my opinion!
Yes, the frees should be tested and when on meds they are a lot more reliable to go by.
Your symptoms may also be related to sluggish adrenals which can affect the thyroid so maybe check in with a naturopath and get this addressed first as in my own experience, adrenal fatigue prevents thyroid meds working properly, and thyroid meds have contraindications stating that they must not be prescribed if adrenal Insufficiency is present.
Also, maybe look at getting iron, vit d3, antibodies for hashi, cortisol testing (min a.m./p.m. But preferably 4 blood draws or 24 hr saliva test)
If you PCP is not willing to fully address this, find yourslelf a thyroid savvy doc who prescribes natural thyroid (dessicated) these are generally the ones who look at the whole picture and not just the TSH. You can go into a compounding pharmacy and ask which docs prescribe compounded thyroid and try to get in and see one! Good luck