Did you switch apple IDs (create a new account) or did you change the one you already had? One of my ID's, the one I use for music, apps, books, etc., it appears I have the ability to change it to whatever but I haven't tried. (That first account I opened up many, many moons ago.) But with the apple ID for iCloud (created the account with the MobileMe software from many moons ago before it was called iCloud), I can only change to the actual ID or to an existing alias of that ID. This is what I learned when I went to appleid.apple.com. Check it out.
What I do know is that I cannot combine the two accounts. With that said, if you're trying to get your iCloud on the new account, it won't work. I've had many Apple associates tell me every time I ask, can I please put it all on one ID? They always tell me it can't be done. I figure it never hurts to ask because they may eventually find a way, right?
"I need to log out of find my iPhone to switch it. How do it get my Apple ID to exist again and then log out of find my iPhone?" <blush> You lost me here! Did you try going to iforgot.apple.com like in your screen cap?
Best of luck to you. 🙂
Mindy