Try any way you can to get one of the doctors to run those tests for you. Regarding another doctor, I am sending a PM with info. To access, just click on your name and then from your personal page click on messages.
From your hormone tests, from my limited knowledge, I see nothing that stands out as a problem.
If you do go to another doctor, it is going to take some time to get an appointment. Do you think you could go ahead and convince your doctor to run those tests now, so you have a better idea of the problems? Don't know if the hormone tests will help or not, but go ahead and post them, with ranges.
Are you restricted to doctors that accept health insurance, or are you willing to pay cash?
First thing is that the T3 Uptake and Free T4 Index tests are very outdated and not very useful when far better tests for Free T4 and Free T3 exist. I would not have much confidence in a doctor that still orders those tests. In addition the doctor blew off the possibility of endocrine problems affecting hair loss, which further lowers his reputation with me. What did the doctor have to say about all those other symptoms you have? What was his response to the nodules? Seems like you need to find a good thyroid doctor, for which we may be able to provide names of some candidates tht have been recommended by other thyroid patients.
In assessing a person for possible hypothyroidism, the most important consideration should always be an evaluation for symptoms that occur more frequently with hypothyroidism than otherwise. You have a number of those. Next is to assess biochemical test results for the biologically active thyroid hormones, Free T4 and Free T3. Your Free T4 of 1.30 is adequate, at about 51% of its range. Your Free T3 of 2.9, however, is only about 28% of its range, which is lower than needed by many people. Note that the range is far too broad to be functional for everybody, due to the erroneous assumptions used to establish those ranges. This results in the ranges being too broad and skewed to the low end. Many of us have found that we needed Free T4 to be at least mid-range, and Free T3 in the upper third of these ranges to relieve hypo symptoms.
Your Free T3 level being much lower in range than you Free T4 is an indication of poor conversion of T4 to T3. There is a conversion process that determines the balance among FT4, FT3, and Reverse T3. FT4 is the main extracellular thyroid hormone, while FT3 is the main intracellular thyroid hormone. T4 is converted to T3 and biologically activated on entry into cells. The conversion process is affected by a number of variables, including levels of iodine, selenium, ferritin, zinc, and even TSH. You should always make sure to test for FT4 and FT3. At next opportunity it would also be a good idea to include Reverse T3 to determine if T4 is being converted to RT3 excessively. Excess RT3 can block the effect of T3, causing hypo-metabolic effects. In addition you need to test for B12 and ferritin and then supplement as needed to optimize. B12 should be in the upper end of its range, and ferritin should be at least 100. Hypothyroid women are especially prone to being low in ferritin, and it is very important for good hair growth.
Hypothyroidism is not just the result of inadequate thyroid hormone. Properly defined, hypothyroidism is insufficient T3 effect in tissue throughout the body due to inadequate supply of, or response to, thyroid hormone. The response to thyroid hormone is significantly affected by Vitamin D and cortisol. Your Vitamin D is lower than recommended so you need to supplement to raise it to at least 50 ng/mL. In addition it would be a good idea to test for cortisol. Either too low or too high can be a problem. So the takeaway from this should be that hypothyroidism is more than just low thyroid hormone levels. Hypothyroidism is insufficient Tissue T3 Effect due to both thyroid hormone levels and the response at the cellular level. Insufficient Tissue T3 Effect causes symptoms. Their is no biochemical test that can determine Tissue T3 Effect. That is why symptoms are the most important indicator of Tissue T3 Effect and thus thyroid status.
As for the nodules, I think the best step would be to find a good thyroid doctor and find out what further testing should be done for those nodules. The nodules are fairly large and may need to be biopsied. If a biopsy is clear, then then question becomes, I think, whether they are obstructing your breathing/ swallowing or having an effect on thyroid hormone output. Following is a good site on nodules.
https://www.mayoclinic.org/diseases-conditions/thyroid-nodules/symptoms-causes/syc-20355262
In summary, I think the first thing needed is to be tested for Free T4, Free T3, Reverse T3, cortisol, B12, and ferritin. With those results available, needed treatment will be more apparent. I know you are low in Free T3, but whether that requires thyroid medication or correction of other variables needs to be evaluated through these tests.
If you want to confirm what I have said, click on my name and then scroll down and read in my journal, the one page overview of a paper on Diagnosis and Treatment of Hypothyroidism: a Patient's Perspective.
In addition you need a good thyroid doctor to do further evaluation for the nodules. I see your location. If you will tell us your insurance carrier perhaps we can locate and suggest a doctor for you to consider.
There is much to discuss, but first please post the reference ranges shown on the lab report for the Free T4 and Free T3. The most important consideration in assessing a person for possible hypothyroidism is symptoms. So please tell us about any symptoms you have.