I think a more fitting term to describe this is being an empath. Sympathising with someone is being apologetic or sad for someone, but empathising with someone is a lot more than that.. you quite literally walk around in the shoes of other people and start to see/feel the world how they do.
While I think this is an amazing gift, the consequence is you're much more likely to feel emotional burnout and overwhelm, which sounds a lot like what you're experiencing. It may be worthwhile for you to lookup "excessive empathy" or "compassion fatigue".
Empathy is a really powerful thing that can benefit a lot of people. For people like yourself and many others, it's very important that you take care of yourself so that it doesn't interfere with your life or burn you out.
I hope I've helped in some way :)