AN LTT; WHILE DEALING WITH A TRIM PROBLEM; FAILED TO TURN TO A NEW VECTOR HDG.

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JUST LEVELING OFF AT CRUISE AT 18000 FT ON A 230 DEG HDG. AS FO ENGAGED AUTOPLT; TRIM BEGAN TO RUN AWAY NOSE DOWN. AT THE SAME TIME; ATC CALLED AND ASSIGNED A 265 DEG HDG. I (CAPT) READ IT BACK; THEN WENT BACK TO DEALING WITH THE TRIM PROBLEM AND RUNNING CHKLIST. AS I WAS DOING SO; ATC CALLED BACK AND VERIFIED OUR HDG. UH OH! I HAD FORGOTTEN OR HAD I EVER REMEMBERED? WE THEN TURNED TO A 265 DEG HDG WHICH TOOK US AWAY FROM THE ARR RTE. GOOD HEADS UP WORK BY ATC. SOMETIMES DURING URGENT DUTIES; WE TEND TO FOCUS ON 1 SITUATION AND LOSE SIGHT OF THE 'BIG PICTURE.' ALTHOUGH; LOOKING BACK; I DO REMEMBER READING BACK THE HDG. I BELIEVE IT WAS DONE SO OUT OF ROTE; SIMPLY TO APPEASE ATC AND LET ME FOCUS ON DEALING WITH THE RUNAWAY TRIM. I'VE BEEN A CAPT NOW FOR 2 MONTHS; AND EACH TRIP I GAIN SOME SORT OF NEW EXPERIENCE. ISN'T THAT WHAT MAKES FLYING GREAT? FLY THE AIRPLANE. KEEP THE BIG PICTURE. THINK AHEAD. DON'T RUSH YOUR THINKING. USE YOUR CREW AS A TEAM. LET THEM HELP YOU. LEAD THE TEAM. READ CALLBACK OF COURSE!