Lots of fibromites have skin problems.
If the sensitive patches of skin are red and scaly, you might have atopic dermatitis (aka eczema). Eczema feels like a sunburn that has blistered and is peeling (looks like it, too).
If the sensitive patches come and go, you might have peripheral neuropathy.
And it could be something else altogether. Ah, the joys of fibro (NOT!).
It's probably worth having a doctor check it out because there might well be something that will either prevent it or improve it.