FLC OF A B737-300 DURING A VISUAL APCH CLBED IN RESPONSE TO A TCASII RA CAUSED BY A B737-500 ON A VISUAL APCH AHEAD AND 300 FT BELOW TO THE SAME RWY. THE FLC STOPPED DSNDING AND CLBED ON 2 SEPARATE OCCASIONS.

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ON APCH TO SEA; SEATTLE APCH ASKED IF WE HAD VISUAL CONTACT WITH ACR X. ACR X B737 AT 12 O'CLOCK POS. WE WERE ON BASE LEG FOR S LNDG; 5000 FT; ABOUT 10 NM N OF ARPT. FO STATED TFC IN SIGHT. ATC INSTRUCTED US TO FOLLOW ACR X CLRED VISUAL APCH RWY 16R. FO READ BACK APCH CLRNC. WHILE DSNDING ON VISUAL APCH CTL ASKED IF WE HAD COMPANY B737 IN SIGHT 2-3 O'CLOCK POS. FO STATED AFFIRMATIVE. NO RESPONSE FROM ATC. WHILE DSNDING THROUGH APPROX 3500 FT MSL OVER BFI ARPT; THERE WAS SEVERAL ACFT ON TCASII. RECEIVED TCASII TA ALERT FOR ACFT AFT AND BELOW US; APPROX 500 FT VERT SEPARATION; ASSUMED IT WAS PATTERN TFC AT BFI. FREQ CONGESTION PREVENTED ATC QUERY. AT APPROX 2500 FT MSL; DSNDING ON VISUAL APCH FOR RWY 16R BEHIND ACR X; RECEIVED TCASII RA TO CLB; WE FOLLOWED ADVISORY; WHEN WARNING ENDED; RESUMED DSCNT TO REGAIN GLIDE PATH. RECEIVED 2ND TCASII RA TO CLB; AGAIN FOLLOWED ADVISORY; NOTIFIED ATC OF RA CLB; HE SAID TO CONTACT SEA TWR. WE LEVELED ACFT AT APPROX 2800 FT MSL SWITCHED TO TWR FREQ. FO CONTACTED SEA TWR; ADVISED WE HAD TCASII RA CLB ON VISUAL FOR RWY 16R. TWR ACKNOWLEDGED US AND STATED NOW CLRED TO LAND ON RWY 16L. FO CONFIRMED THAT APCH CTL HAD CLRED US FOR VISUAL APCH RWY 16R; NOW CLRED TO SIDE-STEP TO RWY 16L; CLRED TO LAND. TWR ACKNOWLEDGED THIS; AND CONFIRMED CLRED TO LAND RWY 16L. WE LANDED UNEVENTFUL ON RWY 16L; AND SAW COMPANY B737 ON PARALLEL APCH RWY 16R AND LOWER; AT 2-3 NM FINAL. TFC WAS WITHIN 300 FT VERT; AND LESS THAN 1/2 NM HORIZ. ATC CONGESTED FREQ ON APCH PREVENTED FIRST ADVISORY OF RA CLB. AFTER LNDG ON RWY 16L; ADVISED TWR OF POOR HANDLING BY APCH CTL AND TWR ACKNOWLEDGED THAT.