B737 FLT CREW REPORTS DESCENDING BELOW GS DURING NIGHT VISUAL APPROACH AND RECEIVING GPWS OBSTACLE WARNING.

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FLYING AS CAPT; MY LEG TO ZZZ; NIGHT VMC; WE WERE VECTORED TO INTERCEPT FINAL FOR RWY. WHEN WE HAD INTERCEPTED AND CONFIRMING RWY IN SIGHT; WE WERE CLRED THE VISUAL APCH; APPROX 20 MI OUT. I DIRECTED THE FO TO SET 2200 FT IN THE ALT WINDOW. THAT FAR ON FINAL; THE ILS SIGNAL WAS 'WOBBLY;' CAUSING THE ACFT TO OSCILLATE BACK AND FORTH; SO I DECIDED TO HAND FLY TO SMOOTH OUT THE FLT PATH. IN SO DOING; I INADVERTENTLY DSNDED BELOW THE FAF ALT; PROBABLY DOWN TO 2000 FT. WE RECEIVED A SINGLE 'CAUTION; OBSTACLE' AURAL WARNING. REALIZING I HAD DSNDED; I ADDED PWR AND CLBED SLIGHTLY. NO FURTHER WARNING WAS RECEIVED; AND I CONTINUED THE APCH TO LNDG. I DIDN'T EVEN REALIZE MY ERROR UNTIL 2 DAYS LATER; WHEN I WAS STUDYING FOR MY PT; AND READ THAT RECEIVING AN ALT WARNING AT NIGHT REQUIRES THE CREW TO GO AROUND; REGARDLESS OF PERCEIVED CLRNC OR CORRECTIVE ACTION. I NOW KNOW THAT I SHOULD HAVE GONE AROUND. FURTHER; I SHOULD HAVE LET THE AUTOPLT MUDDLE THROUGH; OR PERHAPS USED CTL WHEEL STEERING FOR ROLL; TO ENSURE THAT THE ALT HOLD FUNCTION KEPT US FROM DSNDING BELOW THE APPROPRIATE ALT. DURING MY PT JUST 3 DAYS AFTER THIS EVENT; MY INSTRUCTOR DID A GOOD JOB OF EMPHASIZING CREW REQUIREMENTS UPON RECEIVING GPWS ALERTS AND WARNINGS. I THINK THE TRAINING DEPT IS TRYING TO MAKE SURE CREWS KNOW THE RULES.