Well, to be honest, the TSH and antibodies are neither here nor there. They are both just indicators of what's going on - as I explained above. High antibodies may cause symptoms, but I don't think they cause hair loss. And TSH doesn't cause any symptoms at all, whether it's high or low.
So, what you need to know is your actual thyroid hormone levels - FT4 and FT3. They will give you far more information about your state of health. Plus your nutrients, which are bound to be low if you have gut problems. So, you are right, in a way, if you sort out your gut, other things will start to improve. So, how are you getting on with that?
Have you had your vit D, vit B12, folate and ferritin tested? Are you supplementing anything that is suboptimal? Are you doing anything for low stomach acid to increase absorption? If you have low nutrients and low T3, you are going to get all sorts of symptoms - including hair loss. So, forget about the antibodies and the TSH for the time being, and concentrate on those.
Also, are you on a gluten-free diet and taking selenium? Those are the things you need to be doing if you have Hashi's.