Admitting you were wrong is easiest practiced in private first. Say it to yourself, then you’ll manage to say it to them.

Two separate admissions are hiding inside one apology. The first is to yourself, and it is the expensive one, because it means giving up a position you have been defending internally for a while. Once that is done, saying it aloud is mostly logistics, which is why people who cannot apologise have not finished the private part.

Do the first half somewhere nobody can hear you. Say the plain sentence out loud, including the part you have been avoiding. Then go and say it to them within a day, before the admission has time to reorganise itself into an explanation.