how can I save sent emails from the stock Droid Turbo email app to a different email app?
As long as you are syncing your email account and its folders to the phone, you should be able to open and save the same email in the other app. I would think you could see the same info on the other email app as the initial email app given the same info is available in each app because you are using the same email address.
unfortunately the "sent" email (from a POP3 account only seems to be saved in the phone somewhere vs a central server
You can change those settings in the desktop version of your email app's homepage. IE...if you use MSN/Live/Hotmail, go to that desktop site and change your settings to keep the messages on server, or delete from server when deleted from app.