Email app keeps looping with incorrect password message

Yesterday I changed the pw to my gmail account. This morning I tuned on my iPhone 4s and the 'push' for email 'looped' - I got the message that password was incorrect. I first tried to put in the correct(new) password, but the email app kept requesting the pw again and again. So I tried entering cancel to the request, but the app continued to make the request, repeatedly. I could not reach the settings button because the 'request for password" window could not be closed. Phoned AppleCare, and a tech was able to help me through a series of turning off the iPhone and turning back on and - somehow - I got to the settings and was able to delete the email account. I later re-added it.

Then the same dance began with my iPad2. This time I was less successful. When I contacted AppleCare I was told the problem was my devices, the OS, or, basically, PEBCAK - because the email client shouldn't work that way.

Here's the problem, it DID work that way. What made the situation more problematic was that I also hadn't had a chance to change my hotmail pw on my iPad, the app polled my hotmail email account multiple times with the wrong password, often enough that MS security shut down my hotmail account. I had to reset that password now as well.

How do I keep the email app from cycling when there is an incorrect password? A 3rd call to AppleCare yielded the response to turn off push for all my devices, and to manually get email.

Really? No notices for new email?

Since AppleCare yielded two different and equally unsatisfactory answers, I am hoping the community can help.