Is there a way to buy printed pieces to replace stickers? I don't like stickers on my lego models. Does lego sell pieces with custom printed designs, so I can have newer lego models with all printed pieces like the old days? 

If not, how do I print designs on lego pieces?

Every time I look this up on google I get a bunch of results about 3d printing lol.
No, Lego does not sell custom printed bricks in small quantities (I suspect if you wanted thousands of the same design it would be a different story).

To get your search to work you need to use the advanced features, specifically by removing results that contain 3D: print on ABS plastic -3D

That turns up mostly links on screen printing, which would be awfully fiddly to do with Lego pieces.

A friend has a printer that will print properly on Lego, but it was not cheap to buy. IIRC it's an inkjet that takes special cartridges, presumably using the same sort of ink that the screen printers use. If you're willing to spend a couple of thousand dollars on buying a printer I can ask him for details. My solution would be to pay him to print bricks for me, as I also prefer not to use stickers, but I rarely use decorated bricks so that hasn't yet been necessary.