FLC OF AN MLG ACR ACFT TOOK EVASIVE ACTION AND DEVIATED FROM ASSIGNED HDG AND ALT IN RESPONSE TO A TCASII RA AND VISUAL SIGHTING OF THE OTHER ACFT.

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TSTM IN ZNY AREA; WE WERE #2 FOR THE ILS 22 APCH AT LGA. ACR WAS #1. THEY WENT MISSED APCH SOMEWHERE INSIDE OF OM. WE WERE THEN CLRED FOR THE ILS 22 OVER OM; RPTED WIND ON FIELD 300 DEGS AT 62 KTS WITH TSTM. WE THEN EXECUTED MISSED APCH AND RETURNED TO APCH CTL FREQ. WE WERE HDG 180 DEGS AND ISSUED 3000 FT BY APCH CTL. LEVEL 3000 FT; WAS TOLD TO TURN L 040 DEG HDG. WHILE IN THE TURN AT ABOUT A 090 DEG HDG; I NOTE ANOTHER ACFT ON TCASII AT 500 FT ABOVE US AND DSNDING! I START LOOKING VISUALLY FOR THE OTHER ACFT. WE THEN RECEIVE AN RA FROM TCASII TO CLB; WHEN OTHER ACFT WAS ALSO AT 3000 FT. I STARTED CLB AND ALSO COULD SEE VISUALLY ACR AIRLINE PASS OUR 12 O'CLOCK POS. WERE ABLE TO BOTH CLB SLIGHTLY 200-300 FT AND TURN TO R TO ABOUT 120 DEGS TO AVOID MIDAIR. KEEPING ACR ACFT IN SIGHT; SAFE DISTANCE; RETURNED TO 3000 FT AND TURNED TO ASSIGNED HDG OF 040 DEGS. APCH CTLR WAS OVERLOADED WITH TSTMS AND HIGH DENSITY TFC AREA. NO ONE EVER SAID A WORD ON RADIO ABOUT CONFLICT OR TO LOOK FOR TFC. I BELIEVE EITHER CTLR GAVE BOTH ACFT 3000 FT ALT IN ERROR; OR ACR FLT WAS ISSUED 4000 FT; AND DSNDED TO 3000 FT. AFTER GOING MISSED APCH IN A TSTM WITH HVY RAIN; WX RADAR ALL RED PAINTING AND MODERATE TURB; THE LAST THING YOU NEED IS ANOTHER ACFT UP CLOSE AND PERSONAL! THANK GOD WE HAVE TCASII NOWADAYS!