This sounds like a persistence issue, the reason why you see the logon page again and again is that you log on to one server and is then loadbalanced to the other CAS which doesn't recognize the session from the first server so it presents the logon page. You log on and is then loadbalanced to the first server which doesn't recognize the session from server 2 and so on...
Now, I know that Exchange 2013 shouldn't need persistence but I've seen a whole bunch of Exchange implementations that needed Persistence to work just the same. Now in regards to why it stopped working after the upgrade that's a mystery.