Cannot sign out of old Apple ID

I changed my Apple ID (email address) at appleid.apple.com because I will be losing access to my old Apple ID email address. Regrettably, however, when I made the change I was currently signed on to iCloud on my iMac using my old Apple ID.

When I realized this I tried to sign out of my old Apple ID (using System Preferences --> iCloud). While doing so I discovered that the "Find My Mac" iCloud feature was ON, and before I could sign out of my old Apple ID I had to turn off Find My Mac. So I cleared the checkbox for that feature. Then a dialog pops up telling me I have to enter my Apple ID password in order to turn off Find My Mac. No problem, I thought, and I entered my password for my old Apple ID. Fail! The dialog tells me "Your Apple ID or password is incorrect." Of course I repeated this effort several times to be sure I was entering my password correctly (I have no doubt I know my password for the old Apple ID). I also tried the password for my new Apple ID. Nothing worked. The problem, I surmise, is my old Apple, not the password. The old ID is no longer recognized even though System Preferences --> iCloud shows that is the one under which I am signed in.

So, it seems I'm signed in to iCloud under my old Apple ID, but I cannot sign out because I have to enter my password to turn off Find My Mac, and my password does not work because my old Apple ID is no longer recognized.

I've spent several hours over two day searching for solutions on the web, and I see that many others have had the same or a very similar problem, and several solutions have been suggested (which are too numerous and lengthy to repeat here). I've tried all the suggested solutions and none have worked for me.

Help! If you have any ideas I'm eager to try them.

I'm running macOS Sierra 10.12.2 on a late 2009 27" iMac.

By the way, my Apple ID change was successfully updated automatically or manually by me on my iPhone, my iPad, and my Apple TV. Even on my iMac my Apple ID was updated for the App store. But I no longer have access to iCloud on my iMac.