Hypothyroidism. Having any side effects?

When I first diagnosed with hypothyroidism I was prescribed with eltroxin but when I was using that I noticed hyper pigmentation around my mouth and my skin around the mouth got dry. basically mine is a super oily skin but just because of those tablets I got hyper pigmentation around my mouth. my doctor suggested me to use another brand of thyroid medication that is thyronorm but still I am getting the hyperpigmentation around my mouth. In between I stopped the thyroid medication, and my skin was normal. All the hyperpigmentation was gone and my skin was oily again but if I am using those tablets, I am facing dry skin and hyperpigmentation around my mouth. What should I do? Shall I go homeopath. Can homeopathy cure my thyroid issues without having any side effects?