Pressing and holding the play/pause button is how to turn it off...
Just pressing the play/pause button, to play or pause music, works properly and does not cause this problem, correct? It's only when you press and hold the button to turn the iPod off. So, you are able to use your iPod normally, except for turning it off.
If you already did a Restore (from the Restore button on the iPod's Summary tab), I don't have any other suggestion. It must be a hardware problem related to that button, because a Restore should fix any software issue.
However, an iPod that is not touched will automatically turn off the backlight (which uses most of the power) in a few seconds, and go into a power saving mode in a few minutes if it's not playing music. I don't usually use the power off command, because pausing the music and leaving it alone does the same thing.