Celiac and Hashimoto's are usually found together. My daughter and I have both. Gluten is not good for Hashis anyway.
Some people do not convert T4 to T3 and therefore need both. Synthroid made me feel worse. Armour did not work either. I've been taking Tirosint (the purest form of T4) and generic T3 for several years now with great results. You definitely need proper labs drawn and a competent doctor to decipher them.
TSH
Free T4
Free T3
Reverse T3
TgAb
TPO
Always take your T4 an hour away from meals. I take mine first thing in the morning. I take my T3 split (2PM and 6PM)