SWIII ENCOUNTERS WX AND EQUIP PROB AND LANDS ON WRONG; PARALLEL RWY AT RDU.

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AFTER BEING VECTORED TO THE RWY 5R LOC AND CLRED FOR THE APCH; WE INTERCEPTED THE LOC AND GS; CONTACTED THE TWR ON 127.45 (THE ONLY TWR FREQ IN USE). WE WERE CLRED TO LAND ON RWY 5R. WE COMPLETED THE BEFORE LNDG CHKLIST AND I BEGAN READING ALT TO THE DECISION HT. THE PLT STATED HE HAD A NAV FLAG ON HIS HSI AND ASKED IF I WAS SHOWING THE SAME. I CONFIRMED A NAV FLAG ON MY HSI. WE CHKED THE LOC FREQ WAS CORRECT AND IT WAS. AT THIS POINT WE HAD THE ARPT IN SIGHT BUT WERE DISORIENTED IN OUR LOCATION DUE TO THE LOC FAILURE; REDUCED VISIBILITY AND NIGHTTIME CONDITIONS. WE CONTACTED THE TWR AND TOLD THEM OF OUR NAV FLAG TO SEE IF THEY WERE SHOWING A FAILURE OF THE ILS. THE TWR ASKED IF WE HAD THE ARPT IN SIGHT; WE RESPONDED AFFIRMATIVE; AND THEY RESPONDED CLRED TO LAND. WE COMPLETED THE APCH VISUALLY AND LANDED. AFTER TAXIING CLR OF THE RWY; THE TWR TOLD US WE WERE ON THE L SIDE AND TO CONTACT GND CTL. WHAT CAUSED THE PROB WAS THE DISBELIEF THAT WE HAD LOST THE LOC. INITIALLY; WE THOUGHT IT WAS TEMPORARY DUE TO SNOW PLOWS OR GND EQUIP BLOCKING THE LOC ANTENNA BECAUSE THE GS APPEARED TO BE FUNCTIONING. SECONDLY; WE WERE RELYING ON THE TWR TO RPT ANY DISCREPANCY OF THE NAV EQUIP ON THE GND TO US. OUR THIRD THOUGHT WAS THAT WE HAD A TOTAL NAV FAILURE WITH DETERIORATING WX LIMITING OUR OPTIONS TO COMPLETE THE FLT WHICH HAD JUST HAPPENED TO AN ASSOCIATE OF OURS. AT FIRST INDICATION OF THE LOC FAILURE; WE SHOULD HAVE ACTED AND DECLARED MISSED APCH INSTEAD OF TRYING TO TROUBLESHOOT THE NAV INDICATION. THEN WE COULD HAVE TRIED TO DETERMINE THE PROB AND A COURSE OF ACTION TO COMPLETE THE FLT. WE SHOULD HAVE REACTED JUST AS TAUGHT FROM THE BEGINNING OF INST TRAINING AND HOW TO HANDLE SUCH A SIT.