Not sure if this would still be helpful since this thread was from 2 weeks ago, but for reference if you're connecting from AA domestic to CX business/first class on the same day, you have access to Admirals Clubs on the day of travel, which generally have showers. I think they might ask you for your onwards boarding pass but you might be able to show them the Manage My Booking page on CX's site to gain access.

Since you said you'll have been flying for the whole day I assume you're connecting in from an AA domestic flight. If so, when you get off the AA flight you should be able to head to the Admirals Club before exiting the secure side of the terminal.

The same works in reverse, if you're flying CX first/business and connecting to a domestic AA service, you will have Admirals Club access for the remainder of the day and into the next day before 6am. Before the HKG-BOS nonstop I used to connect at JFK and shower either there or upon arrival in BOS.