It may look wrong on the newer machine, but does it produce the A4 output that you are looking for? This sounds like it is just a matter of resolution on the new machine. It is also important to keep in mind that Tableau is not a true WYSIWYG application and you may have to create a layout for each of the display sizes where you are expecting the workbook to be viewed.
The PDF looks 'squished' in the same way as it does on screen.
I've managed to do some more testing now and worked out the issue is to do with text scaling on the hybrid device. They seem to be set to 150%-200% as they run at high resolution and this is causing text to render very differently. In this case making it much larger and so the dashboard doesn't display properly.
Looks like there are other people with similar issues around DPI but I'm not sure yet if there is a viable fix. Telling people to change their screen display options isn't going to work...