Same thing was happening to me. After an hour on the phone with VZW tech support and Samsung tech support they determined that there was no issue at all and everything was working perfectly. Still no email pushing, though. Samsung tech support was especially useless.
The issue issue seems to be with the stock mail app. I downloaded an app called Nine Mail from the play store, removed my exchange account from the accounts settings on the phone, and e-mail and calendar syncing has been flawless ever since. I get emails on my phone before they even show up on my desktop now and no battery issues like on the S5. The app is not free, though on sale right now, and is free to try for a couple of weeks. I'm not shilling for the app, I'm sure there are others too, but the Outlook Preview for Android that Microsoft released is a joke, and this one seemed to work for me and has made me decide to keep the phone when I was planning on shipping it back to VZW earlier in the day because of these ridiculous e-mail issues and the seeming lack of ability to support the product.
Try K-9 Mail. I really like it, and it's free.