ACR X APCH TO WRONG RWY HAD NMAC WITH ACR Y. ACR X TCASII TA RA EVASIVE ACTION TAKEN.

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ACR X WAS BEING RADAR VECTORED FROM SCURRY TO DFW ARPT FOR A LNDG TO THE S RWYS (17 OR 18). UPON TURNING TO A 350 DEG HDG PARALLEL TO RWY 17L AND APPROX 6 MI E OF DFW; WE WERE GIVEN STEP DOWN DSCNTS FROM 11000 FT TO 6000 FT MSL. ABEAM THE APCH END OF RWY 17L WE WERE IN A DSCNT PASSING APPROX 8000 FT AND TOLD THERE WAS A 'HEAVY MD-11' AHEAD THAT WE WOULD BE FOLLOWING TO THE RWY. WE HAD HIM IN SIGHT AND ON TCASII AT APPROX 6 MI. AT NO TIME WERE WE TOLD TO EXPECT A VISUAL TO RWY 18. IF WE WERE TOLD; WE NEVER HEARD IT NOR DID WE ACKNOWLEDGE THE CALL (RADIOS WERE VERY BUSY WITH ARR TFC TO DFW). APPROX 10 MI NE OF DFW ON A 270 DEG HDG WE WERE 'CLRED THE VISUAL BEHIND THE HVY.' NO RWY WAS GIVEN SO WE ASSUMED THE CTLR MEANT TO RWY 17. SHORTLY THEREAFTER; WE WERE TOLD TO CONTACT TWR ON THE FREQ FOR THE W RWY COMPLEX (RWY 18). THIS PUZZLED US BUT RADIOS WERE VERY BUSY; SO WE SWITCHED TO THE TWR FREQ FOR THE E RWYS--ASSUMING THE CTLR MUST HAVE BEEN MISTAKEN. SHORTLY THEREAFTER (8 MI FINAL); WE RECEIVED A TCASII WARNING (TA) FOR TFC THAT WAS 1000 FT BELOW. WE WERE ON A DSCNT FOR THE RWY AT APPROX 3800 FT MSL. THE ACFT BELOW WAS APPARENTLY ON A SIMILAR COURSE BECAUSE AS WE CONTINUED OUR DSCNT; THE ALT SEPARATION WAS QUICKLY DIMINISHING--YET HE WAS STILL DIRECTLY BELOW US. WHEN WE LEVELED OFF; THERE WAS 300 FT OF SEPARATION AND TCASII WAS GIVING THE RA TO CLB. AT THAT MOMENT; TWR TOLD US TO 'TURN R' AND CLRED THE VISUAL TO RWY 18R. ACR Y THAT WAS BELOW US CAUSING THE TCASII WARNING APPARENTLY HAD US IN SIGHT; BUT WE COULD NOT SEE HIM; AND IN THE SECONDS THAT ALL THIS WAS HAPPENING; HE WAS ALSO QUERYING THE TWR ABOUT WHERE WE WERE GOING.