An apology can acknowledge impact without pretending harmful intent existed.

You can say that something you did hurt someone without conceding that you meant it to. Those are separate claims, and refusing to acknowledge the first until the second is granted keeps thousands of arguments running for years. The impact is the part that needs addressing.

Try an apology with no defence in it. I did not realize, I am sorry it landed that way, I will do it differently. Explanations can come later if they still matter, and usually by then they do not.