It sounds like instead of simply signing out of one account and into the other you are somehow managing to try to set up a new account (which Apple will not let you do with an email address it has seen before).
Read this document to learn how to sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac --> Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac - Apple Support
Read this support article to learn how to sign in with your Apple ID on different devices--> Sign in with your Apple ID - Apple Support