HDG TRACK DEV CREATES A PROB FOR ARTCC RADAR CTLR WHO FIELDS THE BALL RATHER NICELY.

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I WAS FLYING THE ACFT USING THE AUTOPLT. A CALL WAS GIVEN BY ATC TO TURN 15 DEGS. THE DIRECTION APPARENTLY WAS R. I THOUGHT I HEARD TO THE L. THE CAPT WAS WORKING THE RADIOS AND HAD TUNED IN DAY ATIS SO HE MISSED MY ERROR. WE WERE TOLD TO LEVEL OFF AT 20000 FT; RIGHT AT 20000 FT; SO WE WENT UP TO 20300 FT BEFORE GOING BACK DOWN TO 20000 FT. OUR PREVIOUS CLRNC WAS TO 22000 FT. MY TURN TO THE L INSTEAD OF TO THE R CAUSED THE NECESSITY TO LEVEL OFF. OUR TCASII ALERTED US AROUND THE SAME TIME; ATC HAD CAUGHT THE POTENTIAL PROB. WE WERE TRACKING ON COURSE WHEN THE 15 DEG TURN WAS GIVEN. THIS COULD HAVE BEEN AVOIDED IF THE CTLR ASKED WHAT OUR HDG WAS THEN ADDED THE 15 DEGS TO IT FOR THE R TURN AND GIVE THE NEW HDG IN 3 DIGITS. I CAN SEE THIS MAY TAKE A LITTLE MORE TIME HOWEVER; THIS COULD HAVE BEEN A NEAR MISS HAD NOT THE CTLR/TCASII BEEN UP TO SPD. SUPPLEMENTAL INFO FROM ACN 304346: I ACKNOWLEDGED AND THEN WAS THINKING THAT'S AN OPPOSITE DIRECTION THAN GOING DIRECT TO APE WHICH WAS OUR FILED RTE AND THEN V-12 MECAN; SO NO REASON TO ASK FOR DIRECT APE. I GOT BUSY LOOKING UP THE ATIS FREQ FOR DAY ASSUMING THE FO WHO WAS FLYING WAS DOING THE 15 DEG R HDG. SHORTLY; THAT BEAUTIFUL TCASII ALERTED US OF 'TFC; TFC;' 20300 FT AND TFC INDICATING ON TCASII ABOUT 4 MI 12 O'CLOCK 700 FT ABOVE WE WERE HDG DOWN TO FL200. WE WERE JUST AT FL200 WHEN HE WENT OVER; 1000 FT ABOVE. BY NOT WATCHING FO FLY HE WAS THINKING L DIRECT APE VOR SO HE TURNED 15 DEGS L. TO CORRECT THIS FROM POSSIBLY HAPPENING AGAIN TO OTHER CREWS; A PROC SHOULD BE DEVELOPED TO 15 DEGS R AND HDG--- ASSIGNED STATED. ALSO INSTEAD OF A HDG FIRST FOR TFC IF THE ALT IS VACANT SUCH AS OURS OF FL200 AND ONCOMING TFC WAS FL210; AN ALT SUCH AS THIS; SHOULD BE CONSIDERED FL200 BEFORE USING THE HDG ASSIGNED.