Changed Apple ID email address. can't log into laptop now

I had to change my email address so I went to appleid.apple.com and made the changes there, verified everything and followed all of the directions. I logged out of my old ID on my phone and iPad and logged in with the new one. My phone now displays all of my text messages as telephone numbers so I have no idea who they are from. My contacts seem to have disappeared after the change.

Another huge issue is my laptop. I log into this laptop using my Apple ID email address but there is nowhere to change that to the new email address. Every time I log in using the old email address, I get pop ups asking me for my FaceTime and iCloud password and when I put in the correct one, it says it is wrong. I tried logging out of the computer and logging into it using the new email address and the password but get the shaking box.

This is worse than moving. Does anyone know how I can fix this? My old email address will be invalid (cable company switch) at some point in early April and I don't want to get completely locked out of my account on my computer. I went to System Preferences, iCloud and can sign out of my apple id there but it said if I do, i have to make another password. I have no idea what that is all about and don't want to lose any links I have to anything by signing out. It is not made clear what happens when you log out of that.