MULTIPLE RWY OP PARALLEL RWYS VISUAL APCH. ACR X HAD LTSS FROM ACR Y AND Z. SYS ERROR.

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THE HDG WAS REFINED TO 180 DEGS ON INITIAL CHK IN ON THE NEW FREQ. OUR POS AT THIS POINT WAS ABOUT THE ATL 080/15. PASSING THROUGH 6000 FT WE WERE TURNED TO A HDG OF 210 DEGS (NO MENTION OF INTERCEPTING THE LOC) AND THE TFC WE WERE TO FOLLOW WAS CALLED OUT TO US TO BE AN MLG AT OUR 2 O'CLOCK POS AT 6000 FT. ALTHOUGH WE HAD NO TFC IN SIGHT AT THE POS CALLED TO US; WE DID HAVE TFC THAT FIT THE DESCRIPTION AT OUR 12 O'CLOCK POS. THE RADIO TFC AT THIS POINT WAS SO HVY WE WERE UNABLE TO EITHER CALL THE TFC OR ARPT IN SIGHT OR QUERY THE CTLR AS TO WHETHER HE WANTED US TO JOIN THE RWY 26R LOC (AS WE WERE RAPIDLY APCHING THE RWY 26R CTRLINE). WE WERE NOW LEVEL AT 5000 FT FLYING THROUGH THE RWY 26R LOC; NEARING THE RWY 27R APCH PATH AND STILL UNABLE TO GET A WORD IN TO APCH CTL. WE KNEW THEY HAD DROPPED THE BUBBLE ON US AND PROBABLY THOUGHT THEY HAD CLRED US FOR THE RWY 26R VISUAL APCH; BUT WE WERE UNABLE TO CONFIRM THIS. WE HELD OUR ASSIGNED HDG AND ALT AS LONG AS WE SAFELY COULD WITH ONCOMING ACFT FOR RWY 27R IN SIGHT. HOWEVER; STILL UNABLE TO CONTACT APCH CTL; WE FINALLY RECEIVED TCASII RA'S ON THE RWY 27R ACFT AND FOLLOWED THOSE RA'S. THE FIRST RA; WHICH WAS ON ACR Y THAT WAS IN SIGHT AND AT OUR ALT; DIRECTED US TO DSND WHICH WE DID. IMMEDIATELY FOLLOWING THAT RA (AND NOW AT 4000 FT) WE RECEIVED ANOTHER RA DIRECTING US TO DSND AGAIN; THIS TIME ON ACR Z THAT WE DID NOT HAVE IN VISUAL CONTACT. WHEN THAT CONFLICT WAS RESOLVED WE WERE APPROX 1 - 1 1/2 MI S OF THE RWY 27R CTRLINE AND AT 3000 FT. AT THAT POINT WE WERE FINALLY ABLE TO BREAK IN TO THE RADIO TFC AND CONTACT APCH CTL. WE TOLD HIM OF OUR POS AND CIRCUMSTANCE AND HE AGAIN ASSIGNED US A 210 DEG HDG! WE QUERIED THAT AND HE CHANGED IT TO 310 DEGS AND 3500 FT TO JOIN THE RWY 26R LOC AND CLRED US FOR A VISUAL APCH TO RWY 26R. AFTER LNDG I CONTACTED THE APCH CTL SUPVR VIA TELEPHONE. I ASKED HIM IF WE HAD MISSED A XMISSION OR NOT FOLLOWED INSTRUCTIONS GIVEN TO US AND HE REPLIED THAT WE HAD NOT MISSED ANYTHING; JUST THAT A 'COUPLE GOT AWAY FROM THE CTLR AND BROKE THROUGH TO THE S SIDE COMPLEX.' ALTHOUGH I THINK TCASII WOULD BE MORE EFFECTIVE IF IT DID NOT REQUIRE THE PLT TO LOOK INSIDE THE COCKPIT TO DETERMINE EXACTLY WHAT IT WANTED HIM TO DO AND THEREBY TAKING HIS EYES AWAY FROM SEARCHING FOR THE CONFLICTING TFC; IT WAS CERTAINLY EFFECTIVE FOR US TODAY IN AVOIDING THE TFC WE WERE FLOWN INTO CONFLICT WITH. IT WAS PARTICULARLY EFFECTIVE CONSIDERING WE WERE FOCUSED ON THE FIRST CONFLICTING ACFT AND NEVER SAW THE SECOND CONFLICTING ACFT.