A PA31 PLT RPTS MISSING THE BUR RWY 8 TXWY C6 EXIT AFTER ATC REQUESTED EXITING ON TXWY C6 BECAUSE OF A DEPARTING B737 ON A XING RWY.

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WHILE THE ACFT WAS ON ITS ROLLOUT ON RWY 8; ATC ISSUED THE EXPECTED INSTRUCTIONS OF 'TURN R ONTO TXWY C6; AND CONTACT GND.' THE CAPT; WHO WAS DESIGNATED FOR THIS REVENUE FLT; WAS FLYING THE ACFT WHILE THE NEWLY QUALIFIED CAPT WAS HANDLING THE RADIOS FROM THE R SEAT (THUS; I WAS ACTING AS SIC). I BEGAN READING BACK 'TURN R ON TXWY C6?' THE REASON FOR THE UNCERTAINTY IN MY VOICE IS BECAUSE I WAS LOOKING FOR TXWY C6 WHICH WE WERE ROLLING PAST AS I READ BACK THE INSTRUCTIONS TO ATC. AS SIC; I SHOULD HAVE TAKEN A FIRM STANCE AND SAID SOMETHING TO THE EFFECT; 'I DON'T SEE TXWY C6' OR 'I BELIEVE WE HAVE PASSED TXWY C6.' DURING THIS TIME; THE OTHER CAPT (PF) DID NOT APPLY BRAKES OR SAY ANYTHING AS I BELIEVE HE WAS ALSO LOST. A FEW SECONDS LATER; AS I WAS SCANNING OUTSIDE; I NOTICED THE RWY '15/33' RED SIGN AND MY ATTN JUMPED. LOOKING IN FRONT OF THE AIRPLANE TO SEE A RAPIDLY APCHING RWY HOLD SHORT LINE (WE WERE NOT ISSUED A LAHSO NOR DOES COMPANY OPS MANUAL ALLOW LAHSO) SO I KNEW WHERE WE WERE. I SAID TO THE CAPT; 'WE'RE ON THE OTHER RWY!' UPON READBACK OF US TURNING AT TXWY C6 TO TWR; I BELIEVE TWR CLRED AN ACR B737 TO TAKE OFF ON THE RWY (RWY 15) WE WERE GOING TO CROSS. ONLY SECONDS AFTER I SAID TO THE CAPT 'WE'RE ON THE OTHER RWY;' TWR IN A HASTY VOICE SAID 'XXX123 STOP!' AND 'XXX123 CANCEL TKOF CLRNC!' THE CAPT OF OUR ACFT PUT ON THE BRAKES HARD AND WE STOPPED WITHIN FT. FROM THE TIME WE PASSED TXWY C6; OUR ACFT WAS MAYBE MOVING AT 20 KTS. TO MY RECOLLECTION; WE NEVER CROSSED RWY 15'S RWY MARKINGS; ONLY THE LAHSO LINE FOR RWY 8. AFTER LOOKING BACK; AS THE PNF HANDLING THE RADIOS; I SHOULD HAVE LET ATC KNOW THAT I DIDN'T KNOW WHERE TXWY C6 WAS INSTEAD OF MAKING ATC GUESS BY THE TONE OF MY VOICE. TO THE CAPT; HE SEEMED SO OBSESSED WITH FLYING THE LOC AND GS PERFECTLY DOWN TO THE RWY (EVEN THOUGH WE WERE ON A VISUAL) THAT WE LANDED VERY FLAT AND PERHAPS LANDED MUCH FURTHER DOWN THE RWY THAN NORMAL TO ALLOW FOR PLENTY OF TIME TO EXIT THE RWY AT TXWY C6 (TXWY C6 IS THE LAST TXWY BEFORE THE INTERSECTING RWY AND IS ALMOST ALWAYS ISSUED UPON LNDG AS A RWY EXIT POINT). TO ATC; THEY MAY HAVE ISSUED TXWY C6 TO US A BIT TOO LATE AND PERHAPS ASSUMED WE WOULD BE PLANNING FOR TXWY C6. THIS; I'M AFRAID TO SAY; IS A CLASSIC RWY INCURSION INCIDENT LIKE MANY I HAVE READ AT NIGHT. I WILL EXERCISE EXTREME VIGILANCE AT NIGHT FOREVER!