B737 WITH ZID AT FL370 EXPERIENCED TCASII TA WHILE BEING VECTORED FOR SPACING; CONFLICTING WITH ANOTHER SAME ALT ACR AND ANOTHER AT FL350.

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FLT TO BWI APPROX 180 MI NW OF PSK VOR; WE RECEIVED A WX RERTE. WE WERE GIVEN VECTORS FOR IN-TRAIL SPACING; TOLD TO REDUCE TO SLOWEST PRACTICAL SPD (220 KTS); AND FLY 190 DEG HDG. AS WE TURNED TO THE 190 DEG HDG; WE NOTICED APPROX 4 OTHER ACFT 30 MI S; ALSO AT FL370 (OUR ALT FL370). APPROX 100 MI W OF PSK; WE WERE GIVEN A 090 DEG HDG. AS WE ROLLED L THROUGH 130 DEG HDG; CTR TURNED US BACK R TO 180 DEGS AND THEN 240 DEG HDG. I QUESTIONED THE 240 DEG HDG AS THERE WAS TCASII TFC IN THAT AREA. THE CTLR VERIFIED THE TURN TO 240 DEGS. AS WE WERE COMPLETING THE TURN TO 240 DEGS; THE CTLR GAVE US A TURN BACK L TO 090 DEGS AND DSCNT TO FL330. THE FO WAS FLYING AND STARTED A L TURN AND DSCNT AT 1000 FPM. WE RECEIVED A TA (TCASII) AS WE DSNDED THROUGH FL368. I NOTED THE TFC WAS APPROX 7 NM AT OUR 2 O'CLOCK POS; CONVERGING. I TOLD THE FO TO INCREASE DSCNT RATE AND TURN RATE. I ACQUIRED THE TFC VISUALLY AND SIMULTANEOUSLY DISCONNECTED THE AUTOPLT; LOWERED THE NOSE (5000 FPM) AND INCREASED BANK ANGLE TO 30 DEGS. I NOTED THE ACFT (ANOTHER CARRIER) PASSED APPROX 1 MI IN FRONT AND 1300 FT ABOVE US. AT THIS SAME TIME; THE CTLR ASKED US TO RPT OUT OF FL330 AND PREVIOUS CLRNC WAS TO DSND AND MAINTAIN FL330. I TOLD THE CTLR; 'YOU HAVE COMPANY ON A 090 DEG HDG AND FL330. WHERE DO YOU WANT US?' THERE WAS A PAUSE BY THE CTLR; FOLLOWED BY A REQUEST TO RPT OUT OF FL350. WE IMMEDIATELY RPTED WE WERE OUT OF FL350. THE CTLR THEN ASSIGNED US 100 DEG HDG AND FL330. AS THIS CTLR HANDED US OFF; HE REPLIED; 'THANKS FOR ALL YOUR HELP BACK THERE.' CONTRIBUTING FACTORS: WX; TFC CONGESTION IN SECTOR; CTLR OVERLOAD; CONFUSING CLRNCS (MULTIPLE).