There are two ways I've ever heard my mother apologize to anyone. The first is to say in a bitterly angry but choked up voice, "Well, I'm sorry I [hyperbolic accusation that no one actually made]," and typically spirals into a rant about how hard she works and how ungrateful everyone is. Example: "Well, I'm sorry I was such a terrible mother that I spent all my MONEY and STRESS and SLEEPLESS NIGHTS on YOU and what you wanted and didn't focus on teaching you to be GRATEFUL for the people who gave you EVERYTHING YOU COULD EVER WANT! Like YOUR MOTHER!"
The other variation is simply to say "sorry" but to say it sarcastically and drawn out into four tonal syllables, sort of like Vietnamese or Chinese. First tone "Sah," second tone lower "ah," third tone even higher than the first and put the stress on this one "AH," fourth tone lower again like you're completely giving up on anyone ever appreciating you "ree."