I'm still not clear on why you don't want to pick them up other than you're pi$$ed at her. Your kids have no part in that.
It sounds as if everything in their lives is up to W.
Maybe you should have a talk with W and say "I want to do my share with the kids. Let's talk about our schedules for the next week and we can plan who will do what. I'm uncomfortable when I have to make changes in my schedule the night before." or whatever your comfort level is.
Would this be a 180 for you?
Me 57/H 58 M36 S 2.5yrs R 12/13
Let me give up the need to know why things happen as they do. I will never know and constant wondering is constant suffering. Caroline Myss
