B757 OVERSHOT ASSIGNED ALT IN ZDC AIRSPACE AND CONFLICTED WITH A D328.

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IN CLBOUT FROM EGE; VMC; CAPT FLYING; COPLT ON RADIOS. GIVEN CLB FROM 15000 FT (ESTIMATED) TO HIGHER ALT; THEN GIVEN HDG. COPLT READS BACK HDG '050 DEGS; FL230;' PLACES IN MCP; CAPT VERIFIES FL230; NO RESPONSE FROM ATC TO CONTRARY. CAPT ENGAGES AUTOPLT IN CLB. CTR CALLS TFC AT 12:30 O'CLOCK POS AND ? MI. CAPT AND FO OUTSIDE SEARCHING (VMC) TFC SIGHTED. TCASII INDICATES TFC AT 800 FT. CAPT AND FO VERIFY VISUAL WITH EACH OTHER. CA[T ASKS FO QUERY ATC ON TFC'S ALT. BEFORE FO CAN XMIT; ACR Y CALLS AND SAYS HE HAS A TARGET AT 6 O'CLOCK POS CLOSING HIS ALT. CAPT DISENGAGES AUTOPLT AND STARTS DSCNT. ATC RESPONDS TO ACR Y. HE'S (ACR X) AT FL220. WE ARE IN DSCNT WITH RA. FO VERIFIES IN SIGHT ACFT PASSES UNDER ACR Y AT FL220 TCASII 'CLR OF CONFLICT' PRIOR TO PASSING. CLOSEST APCH ESTIMATED 300 FT; 3 MI. CAPT AND FO BOTH FELT CLRED TO FL230; IE; EASTERLY HDG; TYPICAL STEP ALT ASSIGNMENT; AND READBACK NOT CHALLENGED. OUR INDICATION OF IMPROPER ALT WAS TA ON TCASII AND ATC'S RESPONSE TO 'HE'S AT FL220.' CREW HAD TFC IN SIGHT AND TOOK COMFORTABLE EVASIVE ACTION MANEUVER TO AVOID TFC; PASSING UNDER AT FL220; AND NOT TO INDUCE A MANEUVER THAT PROVIDED EXCESSIVE G-LOADS TO ACFT AND CABIN. CREW DID NOT LOSE VISUAL CONTACT OR FEEL EXCESSIVE MANEUVERING WAS NECESSARY TO AVOID CONFLICT. I FEEL OUR LAST READBACK OF ALT '230 WITH HDG' VERIFIED OUR ASSIGNED ALT. ADDITIONAL VOICE TFC DURING EVENT PRECLUDED AN EARLIER AND POSSIBLY TIMELY PREVENTION OF SITUATIONAL DEVELOPMENTS PRIOR TO RECEIVING A TCASII RA. SUPPLEMENTAL INFO FROM ACN 496855: OUR CORRECTIVE ACTION WAS NOT SEVERE (NO PAX/CREW DISCOMFORT) AND UPON INITIAL VISUAL CONTACT; VISUAL CONTACT WAS MAINTAINED UNTIL PASSING OF TFC. LENGTHY COM BTWN CONFLICTING TFC AND ATC; LARGELY PRECLUDED OUR ABILITY TO INTERJECT USEFUL COM WITH ATC. FLT ATTENDANT ENTERED FLT DECK DURING INITIAL CRUISE AT 15000 FT. HIGH PERFORMANCE AFT SHOULD BE IMMEDIATELY QUERIED OF ABILITY TO CLB TO HIGHER/PLANNED ALTS AND GIVEN CLRNC TO DO SO TO GET OUT OF HIGHER DENSITY/CLOSE PROX LOWER ALTS. IN THIS EVENT; THE AUTHOR OF THIS RPT WAS NOT GIVEN A SPD ASSIGNMENT WHICH WOULD HAVE MARKEDLY CHANGED (IMPROVED) THE ABILITY TO AVOID THIS SIT AND REDUCE CLOSURE RATES. (WE WERE CLBING AT 305 KTS AND CONFLICTING TFC WAS IN CRUISE AT FL230 WITH SPD UNKNOWN.)