Im currently gluten free and it had helped me in so many ways. I was having all kinds of crazy symptoms out of nowhere, extreme exhaustion, blurry vision, headaches, muscle cramps, heart palpitations, constipation, dizzy spells.. the list goes on.. I was diagnosed with hypothyroidism 4 months ago and immediatley gave up caffeine and eliminated gluten from my diet. -Its not easy to give up most of the foods you eat, but I have felt so much better. I eat very little now compared to before, mostly only fruits and veggies, lots of salad and a shake for breakfast.... I limit my meat intake too to help with digestion and overall health, but I do have fish, shrimp, chicken or turkey 3x a week. -What you put into your body is really important. And for me, being healthy and feeling well, is more important than eating tasty garbage. -Ive also lost 16 pounds so far! And my digestion has never been better. The most important thing I have found is to drink a massive amount of water, dehydration can cause a lot of problems in the body. You should drink your weight in oz, each day... For instance if you weigh 140 lbs, you need 140 oz of water, which is about 17 cups a day. Sounds like a lot, but your body will thank you.
Keep working with your doctor to get your hashimoto's disease under control, as it affects thyroid, if you haven't checked in with him/her recently. As far as gluten goes, most people do not have any problems with it, and it is part of very beneficial foods. If concerned, ask your doctor to test for celiac disease so at least you will know if you have any need to avoid it. .