DEP CTLR CONFUSED ACFT AND ISSUED TURN TO WRONG ONE. THEN SAW CONFLICT AND ISSUED NEW CLRNC.

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OUR FLT DEPARTED XB00 LCL AT DFW. TWR INSTRUCTED US TO CONTACT DEP IN OUR TURN. THE DEP PROC IS MAINTAIN 2000 FT AND HDG 240 DEGS. PASSING 1100 FT MSL THE CAPT (PNF) CONTACTED DEP. DEP IMMEDIATELY STATED OUR FLT #; RADAR CONTACT; TURN L 110 DEGS; MAINTAIN 3000 FT. BOTH MYSELF AND THE CAPT QUESTIONED THE CLRNC AS A L TURN TOWARDS THE E WOULD PUT US BACK TOWARD DEP TFC. THE CAPT QUESTIONED DEP ABOUT THE ASSIGNED HDG AND DEP AGAIN SAID FOR US TO TURN TO HDG 110 DEGS. MY CAPT THEN SUGGESTED I TURN AND CLB AS QUICKLY AS POSSIBLE TO AVOID ANY POSSIBLE TFC. A FEW SECONDS LATER THE DEP CTLR (SOUNDED VERY FLUSTERED) SAID TO US; YOU'RE TURNING L; YOU SHOULD BE IN A R TURN. TURN R HDG 240 DEGS AND MAINTAIN 4000 FT. WE THEN FOLLOWED THOSE INSTRUCTIONS AS EXPEDITIOUSLY AS POSSIBLE. THE CTLR SAID WE HAD 'TAKEN SOMEONE ELSE'S TURN' INTO DEPARTING TFC. THE CAPT ASKED IF THERE HAD BEEN A CONFLICT AND THE CTLR RESPONDED WE WOULD SPEAK ON AN OUTSIDE LINE AND LATER GAVE US A PHONE # TO CALL. AT THE COMPLETION OF AN OTHERWISE UNEVENTFUL FLT; THE CAPT SPOKE WITH THE DEP CTLR'S SUPVR. THE SUPVR SAID THAT WHILE THERE WAS A CONFLICT; THE DEPARTING MD-80 HAD US IN SIGHT. MY CAPT ALSO SAW THE MD-80. I; HOWEVER DID NOT. THE SUPVR WANTED TO KNOW IF WE HAD QUESTIONED THE TURN BACK TOWARD THE DEP COURSE; WE SAID WE HAD AND HE THEN SAID HE WOULD CONTACT THE CAPT AT HOME AFTER HAVING A CHANCE TO REVIEW THE TAPES. WHEN THE SUPVR CALLED THE CAPT LATER; HE SAID THE DEPT CTLR HAD 'CLRLY STATED' OUR FLT # AND 'TURN L HDG 110 DEG.' THE SUPVR ALSO SAID THAT WE WOULD NOT BE VIOLATED CONCERNING THE INCIDENT. HE SAID THE TURN TO 110 DEGS HAD BEEN INTENDED FOR AN ACFT DEPARTING MEECHAM FIELD WHICH WAS HEADED DIRECTLY TOWARD ANOTHER ONCOMING ACFT. I BELIEVE CTLR WORKLOAD WAS A DIRECT FACTOR IN THIS INCIDENT. AT THE TIME OF OUR DEP; THERE WAS A CONTINUOUS STREAM OF JET AND TURBO-PROP DEPS FROM DFW. THIS DEP CTLR WAS WORKING AT LEAST DFW AND MEECHAM AND I DON'T KNOW IF THAT WAS IT OR NOT. IF A SEPARATE CTLR HAD BEEN WORKING MEECHAM; THIS CONFLICT OBVIOUSLY WOULD NOT HAVE OCCURRED. ALSO; EVEN IN THE TURN TO 110 DEGS AFTER HAVING BEEN GIVEN THAT TWICE; I SUGGESTED TO THE CAPT THAT WE VERIFY THE HDG ONE MORE TIME BECAUSE IT SEEMED SO UNUSUAL. WE WERE UNABLE TO DO THAT HOWEVER DUE TO THE CONTINUOUS STREAM OF INSTRUCTIONS AND READBACKS ON FREQ. WE WERE UNABLE TO BREAK IN UNTIL THE CTLR CALLED US WITH THE REVISED HDG AND ALT. ALSO; THE SIT COULD HAVE BEEN MUCH WORSE IF THE MD-80 DIDN'T HAVE US IN SIGHT OR IF CONDITIONS HAD BEEN HARD IMC.