Are we related?
I'm like that! My emotions are sensitive. They always have been, as far back as I can remember. What most people hear about, feel bad about, and then turn away from to go back to their own lives, makes me distressed for days or sometimes longer. Cruelty and suffering can immobilize me at times.
I have to be very careful what I read and what I choose to listen to. Not that I stick my head in a hole and try NOT know what is happening in the world, but I can't permit myself to get overwhelmed. For that reason I do not check the news briefs every hour. There are books and blogs I don't read, and movies I don't watch. I usually know what I can handle and what I can't. And I know where the "off" button is on the TV.
I don't know what your religious faith is, if any, but I always have to turn to the One who knows more than any human being how much awfulness there is in the world and why it's there. And sometimes I have an opportunity to help with the problem that's pressing on my heart - supplies for tornado victims, for instance, or financial help for the little children whose parents have been murdered. Sometimes I handle it just by doing *something* positive, whether it relates to the problem or not; the friendly "Good morning" to my neighbor, or my listening to her, which seems inconsequential, may work in my neighbor's life to offset some burden a bit.
What I can do is only a drop in the bucket, admittedly, but the bucket would be less full without my drop.
Hope this helps a bit.