I wish it were that simple. When I try to enter the password I used for the former Apple ID, it gets rejected. When I try clicking on Forgot Password, it asks if I want E-mail authentication or to answer security questions. I can't use E-mail authentication because the old e-mail server has been shut down. The security question option doesn't work because I get prompted for my birthdate, and when I enter it I get a message saying the information doesn't match their records.
I can log on to iCloud with my new Apple ID and password. But each time I turn my computer on, I get the iCloud logon screen with the old Apple ID indicated by default, prompting me for my password.