Thank you for writing. Everyone doesn’t respond to bad news in exactly the same way. You are, in fact, reacting. Your reaction is to go numb while your brain processes hard information. This isn’t uncommon although it may confuse those around you.
Please give yourself a break. You are not insensitive or uncaring. In fact, you are so sensitive that difficult emotions overwhelm you. By distancing from the feelings at first, you are giving yourself time to figure out how to manage them. All this is probably happening on an unconscious level.
It would be helpful in your relationships with others if you can find a way to say, “Give me a minute to let this all in.” Then withdraw for a bit to give yourself time to deal with your feelings. Be sure to go back to the people who share the loss to let them know that you are also affected and that you are there for them.
I wish you well.