I recommend listening to the Apology podcast. If you've already listened to it, listen again.
I recently apologized to a peer in another department and it went better than I thought. I said, "I apologize. I'm sorry for sending out the wrong info. and creating a confusion on your team." then I went into my corrective action.
My peer liked my corrective action and things seem better. He didn't say 'thank you' or 'no problem', so I'm not sure he accepted my apology. He was just concerned about my corrective action. Next time I can do better with a my "groveling" attitude.