You should not feel bad.
Believe me, you are not the first person to be fooled by police into making a statement or doing something else incriminating. Remember: They are professionals. They are good at what they do. And you are not a professional criminal. Lots of educated, savvy and sophisticated people have fallen for the exact same trick.
Some people may in fact talk police out of arresting them. It is hard for me to know because, typically, I only see the people after they have tried and failed to do this. But you should not be fooled by the suggestion that you can avoid criminal charges by writing a letter of apology to the victim. The letter serves only as your signed confession. And you shouldn’t be taken in by assurances that police just want to hear your side of the story before they decide whom to charge. This decision was made a long time ago. By the time you are sitting with the detective in the interview room, their only purpose in getting you to talk is to build a criminal case against you.
There is only one word you need to say if you find yourself in this situation: Lawyer! I have seen interview recordings in which the detective walks out of the room without saying another thing the moment this word is said.
They may give you a hard time when you do this: What are you trying to hide if you are innocent? But, believe me, deep down they will respect you for it. You have just made their job that much more difficult. Somewhere across town, a prosecutor is looking at the exact same recording as I am. The difference is that the prosecutor is cringing.