A PA34 AT DAL FAILS TO NEGOTIATE THE TURNOFF FROM THE RWY AFTER LNDG RESULTING IN A RWY EXCURSION.

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HAD COMPLETED A NIGHT VISUAL APCH AND LNDG ON RWY 13L AT DAL. ON ROLLOUT; TWR INQUIRED ABOUT OUR DEST ON THE FIELD AND AFTER RECEIVING OUR ANSWER DIRECTED US TO TURN L AT INTXN A2 AND TAXI TO THE RAMP. HAD SLOWED; BUT STILL WAS ABOVE NORMAL TAXI SPD WHEN THE SIGN FOR INTXN A2 APPEARED ON THE L SIDE AND STARTED TURN; FOLLOWING WHAT APPEARED TO BE A 'FOLLOW ME' LINE. THIS TURNED OUT TO BE A HASH MARK ALONG THE SIDE OF THE RWY AND THE ACFT ENDED UP IN THE GRASS APPROX 100 FT SHORT OF THE ACTUAL TURN FOR INTXN A2. ATTEMPTS TO TAXI ON THE GRASS FAILED AND THE ACFT ENDED UP STUCK. RWY 13L WAS CLOSED FOR APPROX 1 1/2 HRS WHILE VERY HELPFUL FBO PERSONNEL PULLED THE ACFT BACK ONTO THE RWY. NO DAMAGE OTHER THAN TO THE PLT'S EGO. FACTORS: NIGHT OPS AT AN UNFAMILIAR ARPT -- ALWAYS A TIME FOR CAUTION. POSSIBLE PLT FATIGUE -- PLT HAD FLOWN APPROX 8 HRS THAT DAY TO REACH DAL. I DID NOT FEEL FATIGUED; BUT IT HAD BEEN A LONG DAY WITH SOME DIFFICULT WX ALONG THE WAY. EASILY MISINTERPED COMBINATION OF SIGNAGE AND RWY SIDE MARKINGS. NOT AN EXCUSE; BUT THE LARGE DISTANCE BTWN THE SIGNAGE FOR THE TURNOFF AND THE ACTUAL TURN; COMBINED WITH THE LOCATION OF THE HASH MARKS (APPROX 40 FT APART SO ONLY ONE WAS CLRLY VISIBLE AS I STARTED MY MISTAKEN TURN); CREATED A SIT IN WHICH A TIRED PLT; EXPECTING TO SEE SOME SORT OF MARKING TO FOLLOW; MISTOOK THE HASH FOR THE TURNOFF MARKINGS. EAGERNESS TO CLR THE ACTIVE QUICKLY. THE ACFT WAS MOVING AT SOMETHING MORE THAN NORMAL TAXI SPD WHEN THE INTXN SIGN WAS PASSED. THERE WAS LITTLE TIME TO EVALUATE THE VISUAL CUES MORE CRITICALLY. CORRECTIVE ACTIONS: FOR ME -- JUST PLAIN BETTER ATTENTIVENESS; LESS EAGERNESS TO QUICKLY CLR THE ACTIVE RWY; RESOLVE TO POSITIVELY IDENT TURNOFFS; ETC; AT UNFAMILIAR ARPTS -- ESPECIALLY AT NIGHT. FOR DAL -- HARD TO SAY; BUT I WAS TOLD BY FBO PERSONNEL THAT THIS HAS HAPPENED BEFORE; INCLUDING A SIMILAR INCIDENT WITHIN THE PAST MONTH. MAYBE NEXT TIME THE RWY MARKINGS ARE REPAINTED SOME THOUGHT SHOULD BE GIVEN TO MODIFYING THE HASH MARKS SO THEY ARE LESS EASILY MISINTERPED. ALSO; THE SIGN FOR INTXN A2 IS PRETTY FAR FROM THE INTXN IT REFERS TO AND RELOCATION CLOSER TO THE INTXN WOULD MAKE IT EASIER TO POSITIVELY IDENT THE INTXN AT NIGHT.