I still want to use the sentence "I'm really sorry" but replace "really" with a bigger word, what can I use?
This gives me an opportunity to define sorry and apology. Also, I'm mentioning this here to make you rethink whether you made some mistake or others? Depending on that, the sentences can be made better and effective.
There is something interesting I read in one of the grammar books. The word "sorry" may be used to express that something really went wrong and you feel sorry. Note that I'm sorry does not always mean that the consequences are only because of you!
"Your kid broke my glass while playing baseball."
"Oh, I'm so sorry for that."
In your case, if you are responsible for the consequences, prefer apology. Because you generally don't apologize for someone else's mistake whereas you can certainly be sorry for that as I said in my example.
My sincere/heartfelt apology. I'll take care that this mustn't happen next time.
However, I don't deny using sorry for the mistake you made. In such case, go with Jim's versions of intensified sorries!
In informal speech, intensification is sometimes accomplished with repetition:
I'm really, really sorry!
Another common version might be
I'm truly sorry!