It is possible to turn the screen off manually while running Navigation. You'll still hear the voice prompts, too. You just have to turn the screen on manually to see where you're going.
As for running other things in the background, pulling data uses a lot of power. Whether it's navigation data or streaming music, it draws a lot of power. Doing both just makes it worse.
I understand your point that it worked fine on your old phone, but maybe you could adapt?