What ws the dosage of T3? If you are having those reactions to such a small dose of even T4, there is likely some problem other than the thyroid med itself that is causing your current symptoms.
Six months ago you had a number of symptoms that are frequently related to hypothyroidism. In addition to that, please post your thyroid related test results and reference ranges from which the doctor diagnosed hypothyroidism. Also, any thyroid related test results and reference ranges shown on the most current lab report. Have you been tested for ferritin and cortisol?
The best test for cortisol is the diurnal saliva cortisol panel of 4 tests at different times of the day. This is for Free cortisol. The usual test at the doctor's office is morning serum cortisol, which is not as useful, but better than nothing.
Both low cortisol and low ferritin can cause reactions when taking thyroid med. It is even recommended that low cortisol be optimized before starting on thyroid med. So I suggest trying to get both of those tested, along with Vitamin D and B12.
If you have the prior cortisol test result and reference range shown on the lab report, please post it. Also if you recall the prior Free T3 result, please post that also.
With your symptoms, and the positive test for Hashi's, along with the relatively low Free T4, the somewhat high Reverse T3, and likely low Free T3, hypothyroidism is indicated. That will require treatment with thyroid med, so it is imperative that you find out about the cortisol and ferritin, and optimize those in order to prevent reactions to thyroid med.
Taking a small amount of T4 will not help. Your system will respond by reducing TSH, thus reducing output of natural thyroid hormone. Serum thyroid is the sum of both natural thyroid hormone and thyroid med, so the net effect will be essentially no change. Only when med dosage basically suppresses TSH will further med dosage increases result in rising FT4 and FT3 levels.
From what you have told us I suspect that you may be too low in cortisol, and maybe also low in ferritin. Either of those will cause reactions to taking thyroid med. So you need to get tested for cortisol and ferritin right away. I highly recommend the diurnal saliva cortisol panel of 4 tests. If the doctor won't do that, or insurance won't cover it, then a morning serum cortisol would be an alternative.
How much Vitamin D do you supplement with daily? D needs to be at least 50 ng/mL. If supplementing B12 is a problem, you should ask about B12 injections. B12 needs to be in the upper end of its range. Ferritin needs to be at least 100.
Do you think you can get this done?