Sorry, not sorry, and other iterations of "i'm sorry." Short, sweet and heartfelt is the most affective form of apology. Too often, people will try and talk the person they are apologizing to out of being upset. They will attempt to justify their prior actions, even in the midst of an "apology." You are not truly contrite if you are justifying midstream, or you haven't come to terms with owning your actions. You can't effectively apologize when you are overly influenced by uncomfortable feelings around the mere act of apologizing, or if unhelpful feelings of guilt are getting in your way.
If you are truly sorry for something you said or did, or the fact that another person is upset or feels bad, SAY SO. If not, DON'T. But don't try to manipulate the situation into something it isn't. Don't go around saying: "You know I didn't mean to do that. You know I didn't mean to hurt you. You are too sensitive. It wasn't really my fault. Well you...blah, blah, blah." The fact is, you did whatever it is you did, and they are justifiably feeling whatever it is they may be feeling. And, it's justifiable based on the sole fact that they are feeling it. So, say you are sorry, or don't, and move on.