Sorry should mean being deeply and honestly regretful over what you have done or said. It should be more of a heartfelt regrets rather than just saying the words.
Here is an acronym of the word "SORRY" that I happened to stumbled upon on my researching deeply about this word. I give the credit to Priyanka .
O- offering apology
R- right from my heart
R- realizing the mistakes and truly apologizing to
Y- you, with a promise not to commit the same mistake again
This should be what is on your heart before you say the word "I'm Sorry."
Apostle Paul said in 2 Corinthians 7:9 "...I am happy, not because you were made sorry, but because your sorrow led you to repentance."