The problem with your button is probably that it's too sensitive.
Normally when the button is
not pushed it should have 0 volts over it, but the transistor inside your module sees a higher voltage resulting in showing the date.
The solution is very simple, solder a resistor from the button to "ground".
In my case a 4.7 Kohm resistor did the trick.
This is also applicable to other modules with the "button always pushed" fault.
This Pulsar had the same problem, a 47Kohm resistor did the trick here.
Opening a led/lcd module:
Carefully take the module out of the case, note it has a sort of spring on the side ( at 4 o clock see my pic.) to preveny it from rotating in the case and to make proper contact with the case.
Remove the three clips that keep the upper and lower half together by pushing them outwards on the
display-side.
Now very carefully pull the clear red side from the bottom side. Note the two clear "pins on the side, be careful not to break them.
Watch out for the bondings on the led display, these are exposed now. Touching them will kill the led display.