I ended up going to the Tower lounge.

They asked to see my *G card downstairs. There was a line to show the card. Then there were the ridiculous elevators. I didn't understand what all you FlyerTalkers were complaining about until I actually experienced them. Really, really ridiculous.

The lounge was indeed packed, and there was indeed a wait for the showers (45mn or so) but it was worth it, the showers were really nice. Immaculately cleaned (even dried off!) between uses. Plenty of towels.

Thanks again everyone for all the advice!

Truly