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Itching can be due to dry skin (xerosis), eczema (dermatitis), psoriasis, scabies, liver disease, kidney failure, iron deficiency anemia, thyroid disease, lymphoma or medications. Please get comprehensive blood work done to rule out these causes. Please treat according to the cause. Symptomatic treatment includes using calamine lotion, Caladryl lotion, antihistaminics. Take oil whole body bath at least three days a week since you already have hypothyroidism it may be the cause.
Hypothyroidism and high BP can be completely reversed without any medicines because they are lifestyle diseases. First, fix your hypothyroidism without medicines naturally with lifestyle and diet. Do the correct basic tests for thyroid function which is seven in number. If you have done only the total T4, total T3 and TSH, it is incorrect. These three tests are commonly done and these tests are incomplete. You are missing the autoimmune cause which should never be left out.
Hypothyroidism does not require lifelong treatment unless you have got your thyroid removed by surgery or has been destroyed by radiation to neck. Thyronorm, Thyrox, Synthroid, Eltroxin which are synthetic T4 may or may not be the correct drug you need. T4 is an inactive hormone, T3 is the active hormone. T4 has to get converted to T3 in the presence of a healthy liver, healthy gut and adequate amounts of certain specific vitamins and minerals. T4 is useless unless it is converted to T3.
Dose adjustment in hypothyroid based on normal standard laboratory TSH ranges is completely wrong. Your laboratory will be normal but you will still have symptoms like fatigue, constipation, hair loss, inability to lose weight, brain fog, etc. Triggers for hypothyroidism include chemicals in your food, water, cooking oils, toilet and kitchen cleaners, stress, infections, cosmetics, etc. Take a consultation and try to most probably reverse your hypothyroidism. Treat the triggers and taper your medicine and you may be able to stop medication eventually.