The solution I use to protect my peace is to not engage with them. I've even stopped defending myself because I know these people are going to believe what they want to believe anyway. And no matter what you do, absolutely nothing will be enough for them. So it's better to do what you want instead of trying to please them.
Admitting your mistakes requires a certain level of maturity. If they had that, they wouldn't have let stuff reach that state of miscommunication. It's okay
I’m in the same situation and honestly I just gave up on my family. They will never change so the best solution is to limit communication with them. I have been doing that recently and it’s been much more peaceful.
Emotional Intelligence is such a curse in a family that does not try to admit mistakes and ignores the truths that you say. They won't listen nor admit mistake even if I beg on my knees for them to hear me out. And I hate that I can't hate them because I understand why they are like that.