If you can pass security and enter the G gates at the international terminal, the SQ lounge should allow you entry and the use of their (single) shower.
We were able to use last week as we were arriving from a domestic flight and simply used the connector walkway. No one mentioned (and we didn't check the screens carefully enough) that OZ use the A gates so had a bit of a mad dash before departure.
Not sure what lounge OZ use around the A gates - assuming you are in Y and that you'll not be entitled to use their contract lounge available for C passengers.
If you have no luck and the SQ lounge deny entry as I hear they have done occasionally in the past, pre-security showers are available near the security area at the entrance to the G gates.
Mark