PF OF B727 FLEW HDG 050 DEGS VICE 005 DEGS AS ASSIGNED BY THE TWR.

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NIGHT FREIGHT. BACKSIDE OF BODY CLOCK; SUBTLE FATIGUE. PF HAS SLIGHT HEAD COLD. DEPARTED RWY 36L AT DFW; PUBLISHED HDG IS 354 DEGS MAGNETIC. TWR HANDS US OFF TO DEP AND ASSIGNS 005 DEG HDG. PF TURNS TO 050 DEGS; WHICH WE FLY FOR APPROX 1 MIN WHILE CAPT (PNF) IS TRYING TO CONTACT DEP. AT ONE POINT; PNF GOES BACK TO TWR FREQ AND ASKS ASSIGNED HDG; WHICH TWR CONFIRMS IS SUPPOSED TO BE 005 DEGS. FINALLY CONTACTED DEP; BUT BY THIS TIME OUR FLT PATH HAS TAKEN US ACROSS AT LEAST EXTENDED CTRLINE OF RWY 36R AT DFW AS WELL. ONLY GOOD THING IS AT XA00; NOT TOO MUCH TFC DEPARTING YET. ON TOP OF IT; FREIGHTERS DO NOT REQUIRE TCASII; SO WE WOULD NOT HAVE SEEN TFC DEPARTING ON PARALLEL IF THERE HAD BEEN SOME. AS FE; I WAS ALSO BUSY MONITORING PANEL; GEAR RETRACTION; AIRSPD AFTER IMMEDIATE LIFTOFF; CLBOUT AND DID NOT HAVE AS MUCH ATTN DEDICATED TO TWR'S HDOF. I DON'T KNOW IF THERE WAS A GOOD REASON FOR TWR TO GIVE US AN 11 DEG HDG CHANGE AND HDOF TO DEP AT SAME TIME; BUT IT IS JUST ONE MORE ITEM TO TRY AND ASSIMILATE AT A VERY BUSY AND CRITICAL PHASE OF FLT.