iCloud not verifying

I changed my iCloud email address on my phone. Then was prompted to put in password on Mac. When it wasn't working after typing it in, I tried deleting iCloud account from computer and reinstating it with the new email address and the old AND new passwords. Now after typing in password too many times it simply says I can't do this now and to try again later. Finally, I've tried to reset my apple ID password (did this through my phone) but its still not working on my Mac with the new AND old email addresses and the new AND old passwords. Incidentally, I was able to log on to this site with new email address and password?!?!? Go figure. What can I do to solve this mess?