Hi
I have had an issue very similar. I changed the email of my Apple ID and the change went smoothly for the most part and my iMac, MacBookPro and iPhone 5 were finally syncing correctly for the most part through iCloud. But on the iMac, Address Book kept requiring the password for my Apple ID with the previous email, which of course was no longer current.
After trying various ideas (such as deleting the previous ID and password from the Keychain access) I finally solved this issue by removing the entire Address Book file out of Library/Application Support and relaunching Address Book. It was empty of course, but it immediately recreates the required files in Application Support. I was then able to sync the "new" Address Book in iCloud via the new version of my Apple ID. (This of course assumes that you have all your contacts on iCloud beforehand.)
Things work fine now but I still think the solution I had to adopt was messy, as indeed was the fact that on my IPhone, I had to completely delete my iCloud account and reopen it with the new email, whereas I expected that since the account wasn't a different one, the change would be automatically and smoothly transmitted.
Plus the fact that to update some apps purchased previously (e.g. Keynote), I had to trash them, and redownload them with the changed Apple ID. You do this for free, but again, it's a messy proces.
I wasn't able to find clear instructions on this anywhere in the info pages about iCloud.
Anyway, to stop your Mac displaying an incorrect ID, you have to sign out of the account, and then sign back in with the correct ID.
You are invited to say whether you want to save your contacts or whatever to your computer or not. Note that if you do save to your Mac, and then let iCloud combine the data when you relaunch, you can end up with a series of doubles…