I have Hashi's, and began perimenopause a bit young a couple years ago. My cortisol levels were a bit off, but not terribly. My gynecologist recommended low dose birth control to help regulate hormonal levels, but I was resistant, thought it would get better, and went a year feeling very uneven. My thyroid med levels also needed to be adjusted over this period of time (which makes sense given the interaction of the HPTA axis.) Six months ago I finally gave in a went on a low dose birth control. I am amazed at how much better I feel. My thyroid appears to also have stabilized.
Your mileage may vary of course, and cortisol testing definitely seems indicated, but that was my experience. The thyroid/adrenals/sex hormones are all very interrelated.