ACR X EMER WX AVOIDANCE NON ADHERENCE TO ATC CLRNC.

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GREENSBORO DEP CLRED US TO CLB TO 10000 FT; THEN GAVE US A HDG POINTING US AT A LEVEL 3 CELL AS SHOWN ON OUR RADAR. WE COULD SEE THE CELL VISUALLY. ITS TOP WAS AT LEAST 25000 FT. APPROX 50 NM SE OF THE CELL; I CALLED GSO DEP TO INFORM THEM WE'D NEED A TURN TOWARD THE R. THEY DID NOT RESPOND -- APPARENTLY BUSY. AT 30 NM FROM THE CELL; I CALLED AGAIN AND SAID WE'D NEED A HIGHER ALT AND A HDG CHANGE FOR WX AHEAD. THEY RESPONDED BY SENDING US OVER TO ZDC AND TELLING US 'UNABLE.' TRIED TO CONTACT ZDC ON THE FREQ GSO APCH GAVE US AND FOUND OUT IT WAS THE WRONG SECTOR. AFTER ABOUT 30 SECONDS HE CAME BACK WITH THE RIGHT FREQ. NOW WE WERE 10 NM FROM THE LEVEL 3 CELL AT 10000 FT MSL. (WE SHOULD HAVE SLOWED DOWN BY THIS POINT BUT THAT HADN'T OCCURRED TO US AS WE KEPT THINKING WE'D MAKE CONTACT WITH A CTLR AND WE'D GET A TURN.) CONTACTED ZDC BY SAYING 'ZDC; THIS IS ACR X AT 10000 FT AND WE NEED A R TURN IMMEDIATELY FOR WX AHEAD.' NO RESPONSE. THE CAPT AND I ARE GETTING PRETTY UPSET BY THIS TIME AND WE AGREE THAT WE ARE ABSOLUTELY NOT FLYING INTO THIS LEVEL 3 CELL WX. ONE LAST TIME I TRIED BY SAYING 'THIS IS ACR X AT 10000 FT AND WE ARE NOW TURNING R TO 360 DEGS.' NO RESPONSE; ALTHOUGH I CAN HEAR THE GUY TALKING TO OTHER ACFT. I KNEW THERE WAS NO PROB WITH OUR RADIOS BECAUSE WE HAD BEEN XMITTING AND RECEIVING OK UP TO THIS POINT. WE TURNED TO OUR SELF-CHOSEN WX AVOIDANCE HDG. A FEW MOMENTS LATER THE CTLR CALLED US AND CLRED US HIGHER. I TOLD HIM OUR HDG WAS FOR WX AVOIDANCE AND THAT WE'D NEED A NEW HDG SOON FOR MORE CELLS AHEAD. BUT HE WAS OBVIOUSLY OVERLOADED. THE SAME SCENARIO REPEATED ITSELF AND WE ENDED UP TAKING A NEW HDG. THIS GUY WAS SO OVERLOADED HE DIDN'T SEEM TO NOTICE. WE SHOULD HAVE DECLARED AN EMER BEFORE WE TURNED -- THAT WOULD'VE GOTTEN HIS ATTN IN BOTH INCIDENTS.