Hi Kate,
You have an emotional type of personality, and you are empathetic.
Those are both gives, and you can be a help to a lot of people.
But there is only one way to not allow the sadness to be addictive and troublesome to you. You must discipline your thinking. The emotions are fueled, driven by the thoughts you think about situations.
Learn ways to stop your thinking over and over. A little thought is fine. Feeling bad is fine for those children. We all do. But being obsessive about it is not good for you.
Get this book, read it, do the exercises in it, stick with it and you will get help. "Ten Days to Self-esteem" by David Burns. (It's a practical do-it-yourself guide to cognitive therapy, good for all of us.)
Meanwhile, an exercise from the book:
When the thoughts occurs.
1. Write them down.
2. Pray for anyone whose hurting. (that's not in the book). Write down your prayer.
3. Next say to yourself. "STOP thinking about this." Get angry, mean it, be determined. Write this down too.
4. Substitute positive thoughts, even though you do not feel like it (the emotion will seek to continue and hang around. You will need to force the words.) Write any good this that could result from the situation. Say the words out loud (when alone). Say them with whatever feeling you can put into them.
Do this every time, as soon as you sense that you are indulging in the sadness too much. There can be a perverse enjoyment of sadness, but that can lead to depression.
Best wishes.