A320 FLC HAS MISCOM WITH ATC WHEN BEING VECTORED FOR RIVER VISUAL APCH. THEY OVERSHOT THE RIVER THEN WERE VECTORED TO THE RIVER AND ASSIGNED THE VISUAL APCH.

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ON APCH TO DCA; WE WERE BEING VECTORED AROUND TSTMS FOR A RIVER VISUAL TO RWY 18. APCH CTL WAS VERY BUSY DUE HVY FRIDAY EVENING TFC THAT WAS BACKED UP DUE TO TSTMS IN THE AREA. AT APPROX 10 MI N OF THE ARPT THE CTLR ASKED US IF WE HAD THE RIVER IN SIGHT. WE RESPONDED THAT WE DID AND HE MUST HAVE CLRED US FOR THE RIVER VISUAL BUT WE NEITHER HEARD NOR ACKNOWLEDGED THIS CLRNC. THE FREQ CONGESTION WAS SO BAD WE COULD NOT GET A WORD IN TO ASK HIM ABOUT A CLRNC AND WE FLEW BEYOND THE RIVER. AT THIS POINT HE ASKED US IF WE WERE FOLLOWING THE RIVER AND WE SAID WE WERE TURNING BACK AND HE CLRED US FOR THE APCH AND TO CONTACT DCA TWR. WHEN WE TALKED TO TWR HE TOLD US TO TURN IMMEDIATELY TO GET BACK OVER THE RIVER; WHICH WE DID. AT THIS POINT WE WERE HIGH ON THE APCH AND HAD TO DSND AT A HIGHER THAN NORMAL DSCNT RATE TO GET DOWN. WE STOPPED ON THE RWY AND MADE THE NORMAL TURN OFF. THE SIT WAS COMPOUNDED BY THE FACT THAT WE HAD A LOW FUEL STATE AND TSTMS WERE MOVING IN FROM THE S WHICH SHUT DOWN THE ARPT SHORTLY AFTER WE LANDED.