From your April results I see that your Free T4 is at 80% of the range, which is higher than the recommended minimum of 50% of the range. Your Free T3 was only at 58 % of the range, which is lower than needed by many hypo patients. Our rule of thumb is FT4 at 50 % min. and FT3 high enough to relieve symptoms.
Your status is complicated further by having a high Reverse T3 level. RT3 is a normal product of conversion of T4 to T3; however, under some conditions excess RT3 is produced, which can block the effect of Free T3. According to several sources, the biological effects of thyroid hormones at the peripheral tissues best reflect the clinical severity of hypothyroidism. Also, sources say that the best measure of tissue thyroid effect is the ratio of Free T3 to Reverse T3, and the ratio should be higher than about 1.8. Your ratio is 1.3, which is way lower than recommended.
Since there are so many variables that can affect conversion of T4 to RT3, it seems the most effective way to address the T3/FT3 problem is to reduce unnecessary T4 to prevent conversion to RT3. Also, to increase the Free T3 level to improve the FT3 to RT3 ratio. I would recommend discussing doing both with your doctor.
I don't know how the doctor came up with the conversion of your 150 mcg of T4 to only 65 mg of the NatureThyroid. The correct converision is one grain of desiccated thyroid med, like NatureThryoid, for each 66 mcg of T4. Accordingly, your new dose should have been at least two grains of NatureThroid. So no wonder you continue to have hypo symptoms.
In my opinion there is no need to switch to yet another med. I think I would go back to the T4 med at a lower dose and increase the T3 med dosage. That would be more like 100 mcg of T4 and 10 mcg of T3 (with increases of T3 as needed to reach the upper end of range and relieve symptoms). Symptom relief should be all important, not just test results.
When you discuss this with your doctor I also suggest that you should ask to be tested for Vitamin D and then supplement to optimize to about 50. Adequate D is very important to a hypo patient.
Unfortunately it seems that going gluten free is being excessively promoted as the answer to many of our health problems. In my opinion I would not worry about going gluten free unless there is medical test evidence that you are being adversely affected by it.