I do say Sorry for many reasons around the concept of courtesy and civility. I will apologize when I feel I have wrong another person. These 2 things are different in my opinion and both are important to my values.

That being said - whenever I "unload" emotionally on someone from a place of sadness (crying and what not), I say I'm sorry. It will just come out, as you said. My T, gives me this look of acceptance and frustration at the same time. She used to ask me what I was sorry for and I would say I don't know. I think I know now, that in those moments, it's for potentially putting a burden on her, dropping/unloading all this energy on her, and now she has to sit with it and dissipate it however she can/does. I'm not sure if I am making sense here and it is still kind of coming together for me as an idea or awareness.