ACR HAS ENG FLAMEOUT AT ALT. USES WRONG CALL SIGN WHEN COM WITH ATC.

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WHILE ENRTE FROM MIA TO CYYZ; WE EXPERIENCED A FLAMEOUT OF THE L ENG AT FL370. IT WAS OBVIOUS THAT WE WOULD NOT BE ABLE TO MAINTAIN THAT ALT; AND I INITIATED A CALL TO ZJX ADVISING THEM THAT WE HAD AN ENG FAILURE AND NEEDED AN IMMEDIATE DSCNT TO A LOWER ALT. HOWEVER; IN THE MIDST OF THE COMMOTION; I USED AN INCORRECT CALL SIGN (DON'T ASK ME HOW I DID IT -- IT WAS WRITTEN ON MY NOTEPAD IN FRONT OF ME AND DIALED IN ON THE ADF). NEEDLESS TO SAY; THIS LED ZJX TO BELIEVE THAT HE HAD AN ACFT ON THE WRONG FREQ. AS JAX WAS CONFUSED; WE HAD TO BEGIN OUR DRIFT-DOWN NOW; AND BEGAN TO DO SO. WE INITIATED OUR DSCNT WHILE ZJX WAS ATTEMPTING TO GIVE THE INCORRECT ACFT THE CORRECT FREQ FOR THE CORRECT SECTOR. WE ALSO CORRECTED OUR CALL SIGN WITH THE CALLOUT OF ALT AND AGAIN STATED THAT WE WERE THE ACFT WITH THE ENG FAILURE AND WERE DSNDING TO A LOWER ALT. ZJX THEN FELT BETTER ABOUT WHO HAD THE PROB AND WHERE HE WAS. THERE WERE NO TFC CONFLICTS; BUT 1 ACFT HAD TO BE VECTORED OUT OF OUR WAY FOR DSCNT INTO A DIVERSION AIRFIELD. AT A LOWER ALT; WE ATTEMPTED A RESTART; WHICH WAS SUCCESSFUL; AND CHANGED OUR DIVERSION FIELD FROM CAE TO ATL. THE REMAINDER OF THE FLT WAS COMPLETED WITH AN UNEVENTFUL APCH AND LNDG WITH 2 ENGS. WE CHANGED ACFT AND CONTINUED CYYZ. THE ONLY WEAK LINK IN THE EVENT WAS THAT MY INITIAL CALL TO ZJX WAS WITH AN INCORRECT CALL SIGN. THIS OCCURRED MOST LIKELY BECAUSE I WAS HOLDING A CHKLIST IN MY HAND AND TRYING TO PULL UP TO DRIFT-DOWN ALT ON THE CRUISE PAGE OF THE FMS. OTHERWISE; CHKLIST PERFORMANCE/COORD; COM WITH THE FLT ATTENDANTS COM AND COORD WITH FLT CTL AND MAINT COORDINATORS WERE TEXTBOOK. SUPPLEMENTAL INFO FROM ACN 283086: THE FLT ATTENDANTS PREPARED THE CABIN FOR LNDG AND BRIEFED THE PAX IN A TIMELY MANNER. MAINT MET THE ACFT IN ATL AND A SUBSTITUTE ACFT WAS PROVIDED.