B737 CREW DEPARTED DCA WITH INCORRECT FLT RELEASE AFTER WX DELAYS AND RTE CHANGES.

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AFTER A LENGTHY DELAY INCLUDING RETURN TO GATE FOR FUEL; WE WERE READY TO DEPART DCA. CLRNC DELIVERY ISSUED US A NEW ROUTING. WE RPTED THIS NEW ROUTING TO DISPATCH. THE DISPATCHER SENT A NEW FLT PLAN AND WE DEPARTED. ONCE AIRBORNE WE NOTICED THAT THE ROUTING THE DISPATCHER HAD SENT WAS NOT THE SAME AS ISSUED BY ATC. WE RECONFIRMED OUR RTE WITH ATC AND WERE TOLD THE SAME ONE THAT WE HAD BEEN TOLD PREVIOUSLY. AFTER SEVERAL REPEATED ATTEMPTS TO EXPLAIN TO THE DISPATCHER WHAT OUR RTE WAS SHE APPARENTLY BECAME FRUSTRATED AND SIMPLY TOLD US THAT WE HAD THE WRONG RTE AND THAT HERS WAS CORRECT. AT THAT POINT SHE ALSO STOPPED COMMUNICATING WITH US. WE NEVER RECEIVED A REVISED FLT PLAN EVEN AFTER WE RPTED A SECOND RERTE WHICH ALSO WAS NOT THE RTE SHE WAS CONVINCED WE WERE ASSIGNED. FUEL WAS NOT A CONCERN AS WE HAD ALREADY ADDED SOME DUE TO THE DELAYS THAT ORD HAD BEEN EXPERIENCING ALL DAY. POSSIBLE MITIGATING FACTORS ARE THAT WE ONLY HAD A MINIMUM LAYOVER THE PREVIOUS EVENING; WE WERE TIRED AS THIS WAS IN ESSENCE OUR FOURTH LEG (DUE TO THE RETURN TO BLOCKS) AND WE WERE RUSHED AS THE TIME BTWN THE ORIGINAL RERTE GIVEN BY CLRNC AND OUR WHEELS-UP TIME WAS VERY SHORT; INCLUDING ENG START; CHKLISTS; RPTING THE RERTE TO DISPATCH AND SAFELY TAXIING TO THE RWY.