A C172 PLT; DURING AFTER LNDG TAXI; MADE A PREMATURE TURN AND TAXIED OFF THE TXWY AT OAK.

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I WAS RETURNING FROM A LCL BAY TOUR IN A C172RG. OAK TWR REQUESTED ME TO LAND ON RWY 33 AND BACK TAXI TO TXWY C. THE END OF THE RWY 33 IS POORLY LIT; AND THE EDGE OF THE TRANSITION FROM RWY 33 TO TXWY C WAS NOT ILLUMINATED. AFTER SEEING A TXWY C SIGN; I COMMENCED MY L TURN ONTO WHAT I THOUGHT WAS TXWY C. AS SOON AS I REALIZED WE WERE IN THE GRAVEL; I RETARDED PWR TO IDLE AND APPLIED BRAKES. BEFORE THE ACFT COULD COME TO A COMPLETE STOP; THE NOSEWHEEL FELL IN A DITCH; AND A PROP STRIKE OCCURRED. I IMMEDIATELY SECURED THE ENG; AND HAD MY 2 PAX EVAC THE ACFT. I SAW AN ARPT VEHICLE DRIVE BY AND GOT HIS ATTN. HE THEN INFORMED GND; FBO (ABC) AND THE FIRE DEPT. AFTER THAT; AN FBO (XYZ) TUG TOWED THE PLANE TO HANGAR. I BELIEVE THAT HAVING THE TRANSITION FROM RWY 33 TO TXWY C UNLIT WAS A MAJOR CONTRIBUTING FACTOR. ANOTHER FACTOR WAS TAXIING ON AN UNFAMILIAR PART OF THE ARPT. MY LNDG LIGHT WAS ALSO OFF DURING THE INCIDENT BECAUSE OF ACFT TAXIING FROM THE END OF RWY 27R; DOWN TXWY C; AND I DID NOT WANT TO BLIND THEM WITH MY LIGHT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR REITERATED THAT THE TRANSITION FROM RWY 33 TO TXWY C IS NOT LIGHTED; IE; NO EDGE LIGHTS OR CTRLINE LIGHTS. HE RECALLS REFLECTORS BEING PRESENT. HE RELATED THERE WAS A TXWY CTRLINE. BECAUSE HE SAW A VEHICLE/ACFT APCHING R TO L; APPARENTLY ON TXWY C; THE RPTR TURNED OFF HIS TAXI AND LNDG LIGHTS WHILE CONTINUING TO TAXI SLOWLY. IN DOING SO; HE LOST PERSPECTIVE OF HIS POS; TURNED L AND PREMATURELY EXITED THE TXWY.