HRT is usually referring to hormone replacement as in reproductive hormones but of course that is what is being done w/ thyroid too ... the 'meds' are replacement for thyroid hormone that your body isnt producing enough of. I do not know of side effects from the meds only severe consequences for NOT taking the meds. You did not say if you are taking them already but:
Are you taking your meds right? Taking 1st thing, on empty stomach, & do not eat (or coffee) for 1/2 hr? Also, if you take calcium, iron or acid reducer, do not take for 3 hrs.
You should ask the doctor to add Cytomel IF the replacement that you are on is Synthroid.
You should also take Selenium, Folic Acid, D3, Zinc, and B Complex.
Hashimoto's Autoimmune Thyroiditis sufferer