Such as hearing about people's personal life problems or watching the news due to all the sad stories in it. Emphasis on the word avoid rather than just be "upset by".
I prefer not to talk politics or share my problems with people who are ____ because most people are empaths who end up feeling sad if they hear any topic of conversation they deem to be negative or unfortunate. It ruins their day, they become sadenned and they say it burdens them.
The words snowflake, empath of sensitive is not suitable.