AZ lounges ("Sala VIP") Borromini (terminal B) and Giotto (terminal C) have showers, but am not sure what lounges does AA subcontract to (I haven't flown them in years). You may want to try asking AA reservations in Italy: +39 06 6605-3169 (Mo-Fr 9 to 17:30 local time; off-hours forwarded to DUB).
Another option is to contact the Hilton, which is connected to FCO, to see if they have any transient shower facilities and at what price.