E145 FLT CREW EXPERIENCES 2 TCAS RA'S WHILE COMPLYING WITH ATC HDG CLRNCS DURING MISSED APCHS TO RWY 19 AT DCA.

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AS WE WERE VECTORED ONTO THE APCH OF VOR/DME RWY 19. WE WERE AT 13 DME FROM DCA VOR AT 4000 FT. AS WE ARRIVED AT THE VDP OF WHAT I CALCULATED FROM DCA VOR; WE WERE STILL APPROX 60 FT ABOVE MDA OF 940 FT AND IN IMC. DURING THE APCH; THE WINDS BTWN 1000 FT AND 4000 FT WERE OUT OF THE SW ABOUT 40 KTS. WE TOLD THE TWR WE WERE GOING MISSED. I EXECUTED A MISSED APCH WITH THE AUTOPLT FOLLOWING THE SOP WHILE CLBING STRAIGHT AHEAD; WE WERE GIVEN A NEW HDG AND A NEW ALT OF 2000 FT. OTHER THAN WHAT THE MISSED APCH WAS. I SELECTED HDG MODE AND DIALED IN THE NEW HDG. WITHIN A FEW SECONDS AFTER THAT; TWR ISSUED 2 ADDITIONAL HDG CHANGES. THE FIRST WAS FURTHER TO THE R TOWARDS THE SW. WHILE IN THE R TURN A NEW HDG WAS ISSUED FOR AN EASTERLY TURN. WHILE WE WERE IN THE TRANSITION FROM A R BANK TO A L BANK; TWR ASKS IF WE WERE ON THE EASTERLY HDG. WE SAID NOT YET. WE WERE IN THE PROCESS OF MAKING THE TURN TO THE L. I DO NOT REMEMBER ANY OF THE EXACT HDGS THAT WERE ISSUED AS I WRITE THIS RPT BECAUSE OF THE HVY WORKLOAD. BUT I KNOW WE RESPONDED TO EACH HDG AND ALT INSTRUCTION. DURING THE MISSED APCH; WHILE MAKING THE TURN BACK TOWARDS THE E; WE SAW 2 OTHER ACFT ON TCAS IN CLOSE PROX AND RESPONDED TO 2 RA'S WHILE IN IMC. DURING THE RA I DISCONNECTED THE AUTOPLT AND HAND FLEW THE RA. WE TOLD ATC ABOUT THE RA. AFTER THE RA; ATC ISSUED US VECTORS FOR THE LDA DME RWY 19 TO DCA. DURING THAT APCH WE FAILED TO BREAK OUT OF THE CLOUDS AND INITIATED ANOTHER MISSED APCH. WHILE ON WITH DEP; THEY ISSUED VECTORS FOR ILS RWY 1. WE BROKE OUT OF THE BASE OF CLOUDS AT ABOUT 500 FT AGL. AT NO TIME DURING BOTH APCHS DID WE TRANSIT THROUGH P-56 PROHIBITED AIRSPACE. ACCORDING TO OUR VOR NAV DURING BOTH APCHS TO RWY 19. WE FOLLOWED THE SOP DURING BOTH MISSED APCHS AND FOLLOWED ALL CHKLISTS. ONE POSSIBLE FACTOR IN THE HDG (POSSIBLE) DEV IS THE STRONG WINDS OUT OF THE SW AT ABOUT 40 KTS. WHICH COULD RESULT IN A DIFFERENCE OF ACTUAL HDG AND GND TRACK OF 10-20 DEGS. THE QUICK HDG CHANGES ATC ISSUED TAKE SEVERAL SECONDS TO COMPLY WITH. ALSO; WHEN FOLLOWING THE RA WE MAY HAVE STOPPED ON OR ROLLED OUT ON A DIFFERENT HDG; FOLLOWED BY RETURNING TO THE LAST ASSIGNED HDG AND ALT.