FLC OF B757 DEPARTS ASSIGNED ALT AS RESULT OF TCASII RA ALERT WHILE SETTING UP FOR APCH AT FLL; FL.

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WE TOOK EVASIVE ACTION IN RESPONSE TO AN RA FROM OUR TCASII WARNING SYS. THE FO WAS THE PF AND I WAS HANDLING THE DUTIES OF THE PNF. THE FLL AND MIA AREA WAS SWAMPED WITH TFC AT THE TIME OF OUR ARR XC20. WE WERE INBOUND ON THE MRLIN 4 ARR TO FLL. THEY WERE LNDG TO THE W ON RWY 27R. ZMA WAS SO BUSY; THE CTLR DID NOT HAND US OVER TO MIA APCH (THEY HANDLE THE FLL TFC) AT THE NORMAL HDOF POINT. FINALLY I WAS ABLE TO GET A WORD IN AND REQUESTED THE MIA APCH FREQ. WHEN I CONTACTED MIA APCH ON FREQ 133.77; WE WERE ABOUT 20 MI FROM THE ARPT. THE CTLR GAVE US A HDG OF 150 DEGS; REDUCE SPD TO 190 KIAS; AND MAINTAIN 7000 FT. WHILE ON THIS HDG AND ALT; THERE WAS AN UNKNOWN THAT TOOK PLACE -- THE CTLR'S VOICE DISAPPEARED AND SOMEONE WHO IDENTED HIMSELF ONLY AS 'TWR' CAME OVER THE RADIO TRYING TO CONTACT AN ACR FLT. THE ORIGINAL APCH CTLR WAS GONE! WE WERE STILL ON FREQ 133.77. WE DID NOT KNOW WHO 'TWR' WAS -- EITHER FLL TWR OR SOME OTHER REMOTE ARPT TWR OPERATOR. THERE WAS OTHER MISC CHATTER WHICH WE DID NOT UNDERSTAND. ABOUT THAT TIME; WE RECEIVED A TA. THE TARGET WAS BELOW US AND CLBING. WE THEN RECEIVED AN RA WITH THE COMMAND 'CLB; CLB.' I GLANCED AT THE RADAR SCOPE AND SAW THE RED TARGET WITH THE ARROW INDICATING HE WAS CLBING. I COMMANDED THE FO TO TAKE IMMEDIATE EVASIVE ACTION CLBING OUT OF 7000 FT. AT ABOUT THE SAME TIME AS WE STARTED OUR EVASIVE ACTION; THE TWR NOTIFIED US HE HAD RECEIVED A TFC CONFLICT WARNING ON HIS SCREEN. I DON'T REMEMBER IF HE CALLED FIRST OR IF I CALLED FIRST; BUT IT SEEMED AS IF WE BOTH RECOGNIZED THE POTENTIAL CONFLICT ABOUT THE SAME TIME. I NOTIFIED HIM THAT WE WERE CLBING IN RESPONSE TO A TCASII WARNING. ABOUT 7700 FT; THE TCASII SAID 'CLR OF CONFLICT.' THE TWR TOLD US TO CONTINUE OUR CLB TO 8000 FT AND TURN L TO A HDG OF 090 DEGS. WE DID NOT SEE THE OTHER ACFT AT ANY TIME -- BEFORE; DURING; OR AFTER THE EVASIVE ACTION. WE ENDED UP ABOUT 30-40 MI E OF THE ARPT. FROM THERE WE WERE ABLE TO MAKE A SUCCESSFUL APCH AND LNDG AT FLL. AFTER WE ARRIVED AT FLL; I HAD THE OPPORTUNITY OF TALKING WITH THE APCH CTL RADAR SUPVR. HE WAS INVESTIGATING THE SIT AND HE INDICATED THERE WAS AN UNAUTH ACFT IN THE AREA.