Not starting with "I am sorry" looks like a good advice - framing matters. Start with description of what happened and how it went sideways, end up with regrett

Not starting with "I am sorry" looks like a good advice - framing matters. Start with description of what happened and how it went sideways, end up with regretting the harm caused - by the time you get there, you'll be perceived as a person who understands the situation and its implications, and is in control of it, and thus will be more inclined to accept the apology and move on.