KING AIR CPR ACFT AT CRUISE LEVEL HAS NMAC WITH B737 DSNDING. CTR NEGLECTED TO HAVE KING AIR FLC CHANGE FREQ AND THE 2 ACFT WERE ON DIFFERENT FREQS. BOTH ACFT TOOK EVASIVE ACTION.

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MY COPLT AND I WERE FLYING ALONG AT FL180 ON OUR WAY TO GSO ABOUT 50 MI NE OF GVE VOR IN A KING AIR BE30 WITH SEVERAL PAX. MY COPLT NOTICED TFC ON THE TCASII APCHING FROM OUR 1 O'CLOCK POS OUT OF FL190. SHE THEN WAS ABLE TO SEE THE TFC VISUALLY AND POINTED OUT A B737. AS THE ACFT GOT CLOSER; I REALIZED I NEEDED TO TAKE EVASIVE ACTION OR HE WAS GOING TO HIT US. I ROLLED INTO A 30 DEG BANK R TURN AND CHANGED HDG BY 45 DEGS. BY OUR ESTIMATES THE B737 GOT AS CLOSE AS 1.5 MI AND 400 FT VERT. ALSO; THE TCASII WAS DIRECTING US TO DSND; WHICH I DID NOT FEEL COMFORTABLE DOING SINCE THE B737 WAS DSNDING TOWARDS US. WE ALSO STATED TO THE CTLR WE TOOK EVASIVE ACTION. WE THEN ASKED THE CTLR WHAT HAPPENED AND HE SAID HE DID NOT TALK TO THE B737; THAT HE WAS BEING HANDLED BY 'NATIONAL' CTR. WE GOT A PHONE NUMBER TO CALL TO TALK TO THE NATIONAL TRACON SUPVR. WHEN WE GOT ON THE GND AT GSO; I CALLED AND SPOKE WITH THE NATIONAL TRACON SUPVR. HE EXPLAINED THAT HE SAW THE WHOLE THING HAPPEN. APPARENTLY; THE CTLR WHO WAS HANDLING US ELECTRONICALLY HANDED US OFF TO NATIONAL CTR; BUT NEVER VERBALLY SWITCHED US TO A DIFFERENT FREQ. THEREFORE; NATIONAL CTR COULD NOT TALK TO US EVEN THOUGH THEY WERE TRYING TO. THE SUPVR ADMITTED THAT ZDC 'SCREWED UP' BY NOT SWITCHING US. ALTHOUGH THIS IS COMFORTING TO KNOW THAT CTLRS ARE HUMAN TOO; IT COULD HAVE BEEN DISASTROUS. I CONTRIBUTE THE CTLR'S WORKLOAD WHICH WAS SUBSTANTIAL AT THE TIME; ALONG WITH A COMPLEX AND CONFUSING ATC SYS AS FACTORS WHICH CAUSED THIS PROB. THE DILIGENT EFFORTS BY BOTH CREWS TO SEE AND AVOID AND EFFECTIVE CREW RESOURCE MGMNT; ALONG WITH FUNCTIONAL TCASII EQUIP ARE; IN MY OPINION; THE COMMANDING ELEMENTS THAT KEPT THIS OCCURRENCE FROM BEING AN ACCIDENT WITH THE LOSS OF LIVES.