I understand because I have similar issues with fearing abandonment (don't think I realized this until the past year or so in therapy) and worrying that people are mad at me or frustrated with me or that I'm annoying them. Lately, this has come out in fear of abandonment by my marriage counselor and my T. I think part of it comes from my parents kind of suppressing their emotions and never arguing, rarely crying in front of me, etc. So it was hard for me to know how they were actually feeling, meaning I had to rely on things like tone of voice, facial expression, etc. And they (both parents) like the few times I made my mom angry or hurt (I was a teenager--it was my job!) were this awful thing that I should feel bad about.
Plus, like you, I'm very sensitive to people's moods and can read that in them. I've recently read that this makes us "highly sensitive people." (There are books about it that I need to read.) The problem is that I can sense, for example, that my husband is in a bad mood and automatically assume it's because of something I did. When often, it's about work or his mother or something else unrelated to me. Or I'll often interpret e-mails or comments in the most negative possible light, like stuff from my friends or my T or marriage counselor. If someone doesn't respond to my e-mail, I figure they don't want to talk to me, when really they were probably just busy.
So, you're not alone in feeling this way. Unfortunately, I can't really say anything to help because this is something I'm working on now too.