It hit me a while back that promptly (i.e. immediately) admitting when I was wrong left the other person unsatisfied. If I'm truly sorry I wait until they've gotten their complaint off their chest. I wait until they've finished and then I apologize.
I don't explain, I don't defend, and I don't apologize before they are finished complaining.
A twist on what many advise, but I find that if I just wait a little, it means a lot to the person who is complaining. If I'm sorry, letting them finish venting before I admit I was wrong and apologize, that brief waiting is little of a price to pay.
And if I'm really sorry, it has hit me that what I do is say "I was wrong, I apologize" followed by (if appropriate) "what can I do to make up for that" or "ok, I'll be doing xyz to make up for that, can you suggest anything else?"
That is what it really means to be sorry, and what it really takes to communicate that I am sorry.