Help! gmail and icloud are fighting on my MacBookPro and iPhone!

I have an old pre-iCloud account that uses the usual icloud.com email address. I also have another iCloud account that is the main one--the one where I pay for extra storage and all that stuff--and this main icloud account is identified by my gmail.com address. All was harmony until about 2 days ago, when I suddenly began getting demands to enter my password, followed by the error message that it was the wrong password. I've spent hours with patient Apple techs on the phone, and the problem seems solved for about 10 minutes, then the same thing begins happening. I'm having my own personal Groundhog's Day on April Fools Day, and I am not fooling about that. It's made worse by the fact that it is also my birthday, and dealing with IT issues is not my idea of a fun birthday. So, wah! Anybody have thoughts?