Would test thyroid antibodies and ferritin. However the itching and rash will also need dermatology follow-up - this may be a primary skin problem and not a thyroid issue. The other thing to consider would be an ANA to screen for other auto-immune issues. Also take a multivitamin, lutein, biotin, zinc and b-complex to support hair/nail health. With poor night vision - check vitamin A level -- if this is low and you are on a normal diet then expore for malabsorption.
What i think might be helpful for you is to see the actual thyroid hormone levels, which would be Free T4 and Free T3. Hopefully the endo you're planning to see will run those. I've seen many folks who were "functionally hypo" (normal TSH, normal Free T4, low Free T3. And of course it appears antibodies can cause hypo problems even when everything is in range.
I sure hope you find someone to help you resolve these hypo symptoms. Hypothyroidism, when not treated in a timely manner, can really wreck your health permanently. Best of luck to you - I've been where you are and I hope you obtain resolution of your symptoms.