I use to be "sensitive" until a mentor gave me this view.

I had the words reversed. I was being insensitive to my emotions.
That is, I didn't like many of my emotions and I didn't want to SENSE them.
I wished I didn't have them. I repressed them.

I'm not particularly religious but this line spoke to me:
"God gave you your emotions, who are you to judge them?"

So I realized the trick it to fully accept all of my emotions, feelings, and bodily sensations.
To look for how they are meant to help me. Easier said then done but I find the effort very worthwhile.
I'm no longer sensitive, or insensitive to my emotions.