AN ACR FLC FLYING AN MD88 AT EWR RPTS A TCASII RA AND NMAC WITH A B737.

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AT 4000 FT; DOWNWIND FOR RWY 22L EWR; HEADING 040 DEGS; CLOSE TO BEING ABEAM TEB; SKY CLR; TALKING TO FIRST NY APCH CTLR. TCASII SET AT 10 MI RANGE; ALL OVER-LIGHTS WERE ON; INCLUDING LNDG LIGHTS. A BUSINESS JET PASSED 3000 FT BELOW US GOING TO TETERBORO. NEXT WE ACQUIRED ON TCASII AND VISUALLY; AN ACFT TO OUR L AT 5000 FT MSL ON WHAT LOOKED LIKE A 090 DEG HDG; CONSTANT BEARING; WITH DECREASING RANGE. I LOOKED BACK AT THE TCASII SCOPE TO VERIFY ITS ALT AT 10000 FT ABOVE US; BUT SOON SAW 900 FT SEPARATION WHICH WE BOTH NOTED VERBALLY. AT 800 FT I TOLD THE FO THAT IT HAD BEGUN TO DSND AND I WAS GOING TO DSND FOR AVOIDANCE. AS I DISCONNECTED THE AUTOPLT AND BEGAN A CTLED DSCNT WE RECEIVED OUR FIRST TCASII ALERT; AN RA OF 'DSND; DSND.' APCH CALLED AT THAT TIME AND WE INFORMED THEM OF OUR INTENTIONS. WE WERE TOLD TO REMAIN AT 3000 FT AND TAKE A 360 DEG HDG. THE OTHER ACFT NEVER TOOK EVASIVE ACTION AND LEVELED AT 4000 FT. LATER LEARNED IT WAS A B737. TRACON IS CALLING THE INCIDENT 'OPERATIONAL CTLR ERROR.' APPARENTLY OUR CTLR FORGOT TO HAND US OFF TO FINAL; AND BECAUSE OUR SYMBOL HAD MERGED WITH THE BUSINESS JET BELOW US; FINAL DID NOT REALIZE WE WERE THERE; AND DSNDED THE B737 TO OUR ALT.