You are not alone. My mind defaults to negative thinking ... I can be speaking about something fairly benign or (dare I) positive, and then immediately think of something negative and go down from there. It has become so routine that I didn't even know I was doing it. I just was barely happy because the negative thoughts would intrude.
I am working on changing the negative thoughts to more positive ones. First step, is realizing I am doing it, and then acknowledge it. And then, think to myself "what would I say to my best friend?" and then try to think of something positive to replace the negative thought.
I also feel quite anxious by some of the negative (hurtful or fear-driven) thoughts (part of my Dx) which cause me to have physical symptoms: nervous stomach, tingling feeling in my arms, my face feels hot, I feel very light headed.
I am sensitive to others, as well. Empathizing with others is a great trait! But when it interferes with how you are feeling (to the point of getting depressed), it is destructive. Internalizing TOO MUCH will cause me to get supremely depressed...so right now, until I am strong enough, I try to avoid as much "triggering" stuff as possible (I am staying away from the mainstream media and journalism right now because I get too upset, scared and triggered).
Have you spoken with your T about this? It's really important to find someone to help you change your brain and how it thinks. I see my brain as a vast network of little rivers and until recently, the little streams and rivers all went into a lake of sadness and depression. I am learning now to try to change my way of thinking so my lake isn't of sadness / depression (I hope that makes sense)
Keep posting. You are not alone. Let us know how you are doing.