According to Izabella Wentz the Thyroid Pharmacist approx 5% with Hashimoto's are coeliac, but over 80% find gluten free diet helps, sometimes significantly. Either due to direct gluten intolerance (no test available) or due to leaky gut and gluten causing molecular mimicry (see Amy Myers link)
Many, many of us test negative for coeliac but are gluten intolerant
You can see thousands of posts on here about gluten free diet helping
What time of day was your blood taken? Your cortisol levels are high & probably needs further investigation. Being on an oestrogen, either contraceptive pill or HRY will cause cortisol to rise. Do you have any other symptoms like rapid weight gain especially around your tummy? Do you have trouble sleeping, acne on your face, high blood pressure? With a cortisol level that high you need to look at why it is high but don't start any supplements until you have been further tested.
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