Man, it's nuts to me how people don't realize that even if the accusation is exaggerated or false (doesn't seem to be the case here), how much better you look if you just own up to it and acknowledge fault?
I've legitimately made jokes towards people that were, in retrospect, offensive, yet they politely didn't show offense or tell me about it. Once I realized I went and apologized and told them I realized my mistake, and specifically that I realized why it was a mistake. Does it really make up for what I did? No, but Jesus Christ, that's how you apologize.
At some point we have to all recognize that we've all done regretful shit in our past. None of us are perfect or free from criticism. But it's on ourselves to own that shit. To recognize the mistake and point out how we're going to fix it.