Are you actually sorry when you say the word sorry?

Recently I have made a few mistakes by not expressing myself clearly. Poor choice of words or lack thereof. For example, I should have used the word “some” instead of “all”. I’m not one who generalizes but it came across that way. My first reaction is to say sorry. Was I truly sorry for what I said? I was sorry for not expressing myself clearly but should I have just said “oops, my bad”?

Is there a difference? Do you say sorry too much?