Tried that. Didn’t help. The old (incorrect) Apple ID was still displayed when I tried to access my iCloud account through system preferences.
Perhaps the solution is to leave iCloud alone for a couple of weeks (or more) and by then Apple may have it working properly. To keep addressing the problem is time-consuming, distracting and, above all, annoying.
I also have the problem that while I have no trouble accessing iCloud webmail I cannot get in using Apple Mail (with the same ID and password, which is rejected.)
Somehow I have managed to survive without iCloud for many years and a few more weeks (if that’s what it takes) won’t make much difference.