Coming clean before you get caught is terrifying and it’s the only version where you keep your integrity.

The version where you are found out and the version where you say it yourself contain the same facts and produce entirely different people. Confessing first is worse in the short run because you lose the chance that it never surfaces. What you keep is the ability to be the one who told the truth, and that is not available afterwards.

If something is sitting there, say it before the deadline that discovery would set. Lead with the fact, without the long explanation of why you did it. Then accept whatever follows without negotiating it down.