You have to be careful what you read... There is really no condition called "adrenal fatigue" that's recognized by the medical community. There "is" a condition called adrenal insufficiency, otherwise known as Addison's disease.
I also can't find any scientific studies, directly linking insulin to the conversion of FT4 to the active FT3. If you can link actual scientific studies, I'd be glad to read them.
The article you linked is not suggesting a high carb diet; it's suggesting moderate carbs, which is about 30% of calories being carbs. It's also suggesting that you speak with a dietician who can help you with this. Carbs should be paired with foods that will slow their conversion to sugar and absorption into the blood stream, as this spikes blood sugar levels, which in turn spikes insulin levels. Over a period of time, this can cause insulin resistance and/or type II diabetes.
Your thyroid antibodies do not indicate that you have Hashimoto's... You should ask for a thyroid ultrasound. Some people have been diagnosed with thyroiditis based on the characteristics of the tissue of their thyroid gland.
High carb diet, such as one that consists of a lot of rice is not good. Rice is a simple carb that breaks down quickly into sugar in your body and if not used right away for energy will goes straight into fat cells to be stored for later use. This is what causes us to gain weight. Heart palpitations can be caused by other things; just eating rice will not cause your thyroid to switch to hyper... in fact, heart palpitations are often present in those of us who are hypo.
The type of carbs you need are complex carbs, such as those found in vegetables - better known a fiber. It isn't necessary to eliminate dairy or gluten from your diet, unless you have a sensitivity or intolerance to them, but you should eliminate sugar and processed foods. If you're going to eat rice, make it brown rice, which is whole grain vs white rice, which is processed.
Are you supplementing vitamin D? Vitamin D deficiency can cause a lot of hypo-like symptoms. What about the low testosterone? That can also cause many adverse symptoms.
Check your iron, vitamin and mineral panel. Your iron or ferritin may be low. Its good that RT3 is low. If it is high you have to get on T3 medicine. Try changing your diet for brain fog. Eliminate sugar, dairy and gluten from your diet, include lots of fresh vegetables and fermented food. Take a good quality multivitamin, fish oil and probiotics. Try it for month, I'm sure you will see the difference.
Your Free T4 and Free T3 are way too low in the ranges, which definitely indicates hypothyroidism.
Rule of thumb is for FT4 to be about mid range; yours is only at 24% of the range. Rule of thumb for FT3 is upper half of range and yours is only at 25% of range, so you have a long way to go.
It's hard to say exactly what's going on... what were the exact levels of your antibodies, do you know? It's possible they just haven't ramped up or you could have Central hypothyroidism, which is a pituitary/hypothalamus issue, not a thyroid problem.
Hypothyroidism is hypothyroidism and needs to be treated with replacement thyroid hormones, whatever the cause.
Free T4 is converted to either Free T3 or Reverse T3. With your Free T4 as low as it is, most likely, it's being converted to Free T3, which is what your body uses. rT3 is inert.