B737-300 MAKING LOC APCH TO RWY 36R AT TUL MAKES GAR AFTER RECEIVING EGPWS OBSTACLE WARNING.

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AFTER ROLLING OUT ON A 10 MI FINAL FOR WHAT WAS GOING TO BE THE ILS RWY 36R AT TULSA; WE BOTH NOTICED OFF FLAGS FOR THE GS. WHEN THE CAPT ASKED APCH ABOUT THIS; THEY SAID THEY HAD BEEN HAVING PROBS WITH IT LATELY. WE QUICKLY BRIEFED UP THE NECESSARY ITEMS FOR THE LOC RWY 36R; COMPANY PROCS WITH MYSELF ACCOMPLISHING THE LNDG. VISIBILITY WAS AT THE APCH MINIMUMS WITH MODERATE SNOWFALL. AFTER THE FAF; I BEGAN A DSCNT TO THE STEP-DOWN ALT OF 1300 FT MSL (MDA WAS 1060 FT). I FORGOT ABOUT THE 1000 FPM DSCNT RATE RESTR BELOW 1000 FT AND CONTINUED TO DSND TO THE STEP-DOWN ALT AT 1200 FPM. AS I BEGAN TO LEVELOFF; PASSING THROUGH ABOUT 1500 FT MSL; WE RECEIVED AN 'OBSTACLE; OBSTACLE' CAUTION FROM THE GPWS. SINCE WE WERE IMC I BEGAN TO ADVANCE THE THRUST LEVERS TO GO AROUND WHEN WE RECEIVED A 'PULL UP; PULL UP' WARNING. AT THIS POINT I ACCOMPLISHED THE TERRAIN AVOIDANCE MANEUVER. I CLBED ABOUT 1500 FT; NO FURTHER WARNINGS WERE RECEIVED SO I REDUCED MY PITCH AND CONTINUED WITH NORMAL GAR PROCS. AT THIS POINT WE HAD NO CONTINGENCY FUEL LEFT FOR ANOTHER APCH SO WE DIVERTED TO MCI. AFTER LNDG; WE TOOK A HARD LOOK AT THE APCH PLATE; THINKING MAYBE WE HAD MADE A MAJOR MISTAKE (IE; WRONG ALTIMETER SETTINGS; MISREAD ALT RESTRS) BUT COULD FIND NONE. HOWEVER; WE DID FIND THE OBSTACLE. IT WAS A TWR OF 954 FT MSL JUST BELOW AND TO THE R OF THE FINAL APCH COURSE. ON OUR SECOND ATTEMPT INTO TULSA ABOUT 2 HRS LATER; I FLEW THE SAME APCH; BUT USED ABOUT A 700 FPM DSCNT RATE SO AS TO ARRIVE AT A POINT JUST PRIOR TO THE STEP-DOWN FIX AT 1300 FT MSL. WE ALSO WONDERED WHY THERE WAS ABOUT A 2 SECOND GAP BTWN THE 2 CALLOUTS MADE BY THE GPWS. WE WEREN'T SURE WHETHER WE HAD RECEIVED AN OBSTACLE CAUTION AND THEN AN OBSTACLE WARNING; OR WHETHER IT WAS A WARNING WITH A LONG DELAY IN THE MIDDLE. THIS WAS MY FIRST TIME HEARING THE GPWS SPEAK IN FLT; SO I'M NOT SURE IF THAT IS NORMAL TO HAVE A DELAY BTWN THE 2 STATEMENTS. FIRST; IT'S POSSIBLE THAT MY DSCNT RATE OF 1200 FPM BELOW 1000 FT WAS JUST ENOUGH OF A CLOSURE RATE TO SET OFF THE GPWS. SECOND; WHEN BRIEFING A NON-PRECISION APCH; LOOK AT THE OBSTACLES THAT ARE CAUSING YOU TO HAVE AN ALT RESTR BEFORE DSNDING TO THE MDA. THIRD; LOOK AT THE DSCNT RATE NECESSARY TO ARRIVE AT YOUR STEP-DOWN ALT IN A TIMELY MANNER. IT MAY BE MUCH LESS THAN OUR USUAL 1200 FPM. SUPPLEMENTAL INFO FROM ACN 719052: I WAS FOCUSED ON MAKING THE STEP-DOWN FIX AND DO NOT RECALL VERBALLY CALLING THE 1000 FT CALL.