One of the things I learned from the time I did with Hashimoto's and hypothyroidism, is that the same event (leaky gut, or gluten enteropathy in my case) which causes hypothyroid, also leads to an inability to efficiently digest protein. So at the very time you are feeling horrible from endocrine dropout, you can find yourself also the target of various opportunistic bacteria and viruses that make life miserable, because your immune system can't fight them.
I learned this from experience. I started using free essential amino acids due to sleep issues, and that was a big help. But after several years of this, I also noticed that I was free of the annoying opportunistic illnesses that I had had problems with throughout my life.
Why? Immunoglobulins (e.g. IgA) require amino acids to be manufactured in the body. There are many websites that tell you about the immune-amino connection, such as aminoacidstudies.org/immune... . But you probably will not learn it from your doctor.
The simplest indicator of what is going on in your body, is to get a Total Blood Protein test. The normal range is about 6.2-8.3 g/dl. And the lower part of that range (just as with FT3) is suspect; my doctor, upon request, researched his patient records and told me his healthy and vigorous patients all have Total Blood Protein in the upper half of that range.