While I am not an expert on thyroid.... I do have some suggestions based on your test results.

T4 has 4 iodine molecules on it, and T3 has 3. T4 is mostly made by the thyroid gland and is inactive. T3 is also made in small amounts and will respond to stress/injury/trauma hopefully. If you do not consume iodine in your salt (say you are on a low salt diet) or you do not use iodized salt, or you do not use a multivit with iodine in it, then you should consider using a kelp tablet daily. The average suggested iodine dose is 150-200mcg a day.
Here is a monograph on Iodine:
http://lpi.oregonstate.edu/mic/minerals/iodine

T4 is distributed to the tissues and converted there to T3 as needed. This is supported by two minerals, zinc and selenium.
So if you are low in either or both, you won't convert well.

Make sure you are taking a good multivitamin.... Centrum Silver will have all 3 nutrients in it.

Another thing to consider is that T3 is the cofactor for conversion and activation of betacarotene (pro-vitamin A). If a person is low in T3, or not converting, or any other hypothyroid reason, then you will not be making real Vit A for your needs.
When a person is an extreme hypo...their skin can turn orange...with the build up of betacarotene. But not everyone does. Centrum has a mixture of Vit A and betacarotene. It has 2500IU combined, with betacarotene listed at 40% of that.

When I was hypo years ago I had the orange skin. It went away slowly as I was corrected with T4.

Today I use 8000IU of Vitamin A acetate , to supplement the A separately because multivits don't have enough. I had a noticeable improvement in dry eyes, and skin after only a month on this. Once I seemed so much better I dropped from one a day to one tablet 3 times a week. I bought my Vit A at Target. It was very inexpensive. One should not increase doses of A....past the RDA. Vit A is toxic in long term high doses (upper limit is 10,000 IU) but it is critical for hypothyroid patients IMO. Just use it carefully.
http://ods.od.nih.gov/factsheets/VitaminA-Consumer/

I think getting tested for Hashimoto's is a good idea. But not all hypothyroid issues are autoimmune. The thyroid is also in a partnership with adrenal functions. This is why going to an endocrinologist is a very good idea. This type of doctor can evaluate both adrenal issues and the thyroid issues.

http://www.holtorfmed.com/adrenal-he...id-connection/