Both are correct.

Moreno/a can mean dark-skinned, dark-haired, tanned, brown (as in sugar or bread) etc So in answer to your question, both are correct

As others have said, it can mean both, depending on the context. I would like to add that you can specify saying "morena de pelo" or "morena de piel". If you don't specify or there is no context, I can tell you that most people would assume that you are referring to the skin. In any case, for brown hair, I myself would always say "castaño". So for "I am brown haired" > "Mi pelo es castaño"

It can be both, referred to hair and skin. We also say: "estoy moreno/a" with the verb "estar" after some days sunbathing as the sking changes its color and it turns darker. Many people in Spain like sunbathing and "están morenos en septiembre".

Moreno o morena: Es un gentilicio para referirse a una persona de color de piel de mezcla entre negro y blanco, es un color de piel características del caribe.