You may be overdue for a blood panel (measurements of 3 or 4 different thyroid indicators), follow-ups usually just measure one of the indicators.
Plain dry skin ought to react well to regular use of skincare lotion after bathing.
Dry skin and hair loss are symptoms of hypothyroidism, as are fatigue and memory issues. You should get tested for the thyroid hormones, Free T3 and Free T4, along with TSH, which is the pituitary hormone that stimulates the thyroid to produce its hormones.
You should also get tested for Hashimoto's, which is an autoimmune thyroid condition and for which symptoms can appear long before Free T3, Free T4 and TSH go out of range enough to show a problem. Those tests are Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb). You need them both because some of have one or the other and some of us have them both.
Iron deficiency can also cause hair loss and iron is also necessary for the proper metabolism of thyroid hormones, so you should get tested for ferritin, which is the iron storage hormone.
Biotin, which is a B vitamin might also help stop your hair loss. Ask to get tested for vitamin B-12 as deficiency can cause severe fatigue.