Sometimes when we say the we are sorry, we do not mean it as an apology but rather as sympathy for a person's situation. When someone suffers a loss or has a problem we say we are "sorry" because we know they are sad and we want to convey our sympathy. I think the intent gets blurred sometimes.
If you are always apologizing for your actions and you want to change that maybe just change your words. Instead of, "I'm sorry" use something like, "didn't mean to do that" or something more casual. The ever popular, "My bad" or something along those lines. I am sure there are plenty of other casual terms swirling around. It takes the formality off of it, making it a less apologetic sounding.