Re: hypothyroidism and irregular menstrual cycles!
I would suggest getting cortisol levels checked, being starled easily and anxiety etc while could be related to thyroid are also to my knowledge strongly suggestive of adrenals struggling. Treating adrenals will assist your thyroid treatment.
Your TSH at 3.8 is still quite high, most people with normal function have a TSH closer to 1. Getting your frees tested is better than the TSH once you are on treatment but your TSH is still too high for someone taking meds. Frees are FT4 and FT3.
It might also be worth checking iron, ferritin, iron saturation, vit D3 along with cortisol for adrenals, these are generally low in people with thyroid issues.
The thyroid and the adrenals can all affect your cycle, cortisol is the precursor for most of our hormones, oestrogen progesterone testosterone thyroid hormones and so on, without correct cortisol all of these can be out of whack. Even low normal cortisol can affect this. If it comes back in the low range (e.g less than 60% of the lab range given) doctors wont look at it, then look at a good naturopath and perhaps acupuncture to get adrenal support, helping the adrenals will also assist with the thyroid issues.
Hope you get some improvements soon.
All the best.