“Sorry you feel that way” apologizes to their feelings and protects your behavior. Try the other order.

Apologizing for their reaction quietly relocates the problem. It treats their reaction as the thing that went wrong, which leaves your part untouched and hands them the job of being less upset. People hear the move even when they cannot name it, and what lands is that you would rather be right than repaired.

Put the action first: I did this, it was not okay, I am sorry. Their feelings can come second, as something you acknowledge rather than something you apologize for. If you do not think you did anything wrong, say that honestly instead, because a disguised defense is worse than an open disagreement.