Q: I changed my Apple ID but didn't change everwhere

I have recently changed my Apple ID, I did this on my iPhone since it didn't quite work out on the Mac (it asked me to change my password before I even could log in). After I've changed it, it work as it should until I found out that it hadn't changed the Apple ID anywhere else than the Apple Store. So I logged out and logged in again, with the new Apple ID. This can't be true that you need to do this on all your devices. I have a Mac, iPad, iPhone and a iPod, and each device has several Apple apps, which needs the Apple ID: FaceTime, iCloud, iTunes, Find My iPhone, iBooks, etc.

This either needs fixing or someone tell me how to do it properly. Apple support community