You arrive in T3 and leave from T3. The CX lounge happens to have very nice showers -- better than the AA Arrivals lounge -- but might not be open when your AA flight arrives (they do open about 4 hours prior to the CX flight departure times, so it would be close). Getting to the AA Arrivals lounge involves going landside through customs and is probably not worth doing. Going to another terminal to use a lounge is technically not allowed (sometimes it works but usually does not). In your circumstance it really isn't worth the hassle, use the CX lounge. It has slightly better food/drink than the Terraces or the FL and much faster internet terminals (also more internet terminals than all the AA and BA lounges put together; not quite true, but close). The CX lounge is by gate 1 in T3 (a bit out of the way).