RWY INCURSION; UNAUTH RWY ENTRY OP XING.

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IT WAS MY SECOND TIME IN STL AND THE FIRST TIME TO TAXI TO RWY 12L. THE EARLY MORNING WAS STILL DARK WITH AN INDEFINITE CEILING AND VISIBILITY AT 7/8 OF A MI WITH LIGHT RAIN. THERE WAS NO LNDG TFC AT THE EARLY HR AND ONE OTHER DEPARTING TURBOJET; WHO WAS DIRECTED TO FOLLOW ME. WE WERE CLRED TO TAXI FROM THE RAMP TO RWY 12L VIA TXWY E. IMMEDIATELY AFTER XING RWY 12R AT TXWY E; TWR CLRED US FOR TKOF ON RWY 12L. THE COPLT WAS ACCOMPLISHING THE BEFORE TKOF CHKLIST AS WE APCHED THE INTXN OF TXWYS E; C; AND P. I COULD SEE RWY 12L; BUT IN THE DARK; RAIN; AND LOW VISIBILITY; IT APPEARED TO EXTEND PAST THE INTXN. IN ORDER TO GET THE FULL LENGTH FOR TKOF; I COULD TURN SLIGHTLY L AND TAKE THE NEXT R. THAT COURSE OF ACTION PUT US ON RWY 6/24; WHICH WAS AN ACTIVE RWY WITH NO TFC. TWR WAS NOTIFIED AND THEY DIRECTED US TO TAXI DOWN RWY 6 TO RWY 12R FOR TKOF; WHICH WE DID. REVIEWING WHAT HAPPENED; THE VISUAL CUE OF THE SHORT OVERRUN; OR TXWY; CONNECTING RWY 12L TO RWY 6/24 AND MY UNFAMILIARITY WITH THE ARPT CONTRIBUTED TO THE CONFUSION. ONE POSSIBLE RECOMMENDATION IS THAT IN LOW VISIBILITY; ACFT USE TXWY C FROM THE RAMP AREA INSTEAD OF TXWY E WHEN RWY 12L IS IN USE. AS A SIDE NOTE: THIS DEP REPRESENTED A RADICAL CHANGE IN BODY CLOCK; WITH THE PREVIOUS DEPS AT THE FIRST FLT OF THE WORK DAY COMING AT XA00 EASTERN AND XA39 EASTERN.