You should always make sure they test for Free T3, not Total T3. Only a very small portion of Total T3 is free of protein and thus biologically active. That small portion is called Free T3. Free T4 is not the only thing that matters, Free T3 is more important.
Your thyroid status results from the extent of T3 effect in tissue throughout your body. So you always need to know your Free T3 level along with Free T4. TSH is not all that important, especially when taking thyroid medication. Your FT4 is at 87% of its range, while your Total T3 is at 38% of its range. It is typical when taking T4 med that T3 lags behind T4. This is due to reduced conversion of T4 to T3. So with T4 only med, in order to get T3 high enough it requires T4 to be in the high end of the range, or above. You really don't need T4 above mid-range, so you could reduce your T4 med a bit, if the doctor was agreeable to adding some T3 to your med, to get it into the upper half of its range, and adjusted as needed to relieve symptoms. Symptom relief should be all important, not just test results.
So I have questions for you. First, did you take your T4 med the morning before the blood draw for those tests? If so, how long afterward was the blood draw? Next,and most important, what symptoms do you have currently?
Free T3 does all the work in your body. It is like gasoline in your engine. T3 is used by almost the entire body. T4 is only a prohormone. To be utilized it has to be converted to T3. If you need info to use to persuade your doctor, click on my name and then scroll down to my Journal and read at least the one page Overview of a paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective. There is also a link to the full paper. You can use any of that with your doctor to get him to understand the importance of FT3 and that your thyroid status is dependent on having adequate TISSUE T3 EFFECT.
In addition, hypo patients are frequently deficient in Vitamin D, B12 and ferritin. If not tested fro those you should do so and then supplement as needed to optimize. D should be at least 50 ng/mL, B12 in the upper part of its range and ferritin should be at least 100.
A high FT4 would be a cause for anxiety only if it was being converted to too much T3 and your FT3 was too high.
You should always defer your morning dose of thyroid med until after the blood draw, to avoid false high results. The T4 med causes a short term spike in your T4 level and less so in your T3 levels. Even the ATA/AACE Guidelines suggest waiting until after the blood draw to take your morning dose.
If you can swing it, it would be worthwhile to also get a Reverse T3 done at the same time s the FT3. That way you can calculate the ratio of FT3 to RT3.