LESS THAN STANDARD SEPARATION BETWEEN TWO ACR ACFT. OPERATIONAL ERROR.

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WE WERE INBOUND TO DFW. DELAYS ON ARRIVAL INCLUDING HOLDING; VECTORS AND SEVERAL SPEED CHANGES. FLT WITH SIMILAR FLT NUMBER ON FREQ. WE WERE GIVEN DESCENT TO 8000' FROM 11000' BY DFW APCH CTLR AFTER PASSING SCURRY VOR ON SCURRY NINE ARRIVAL. AT 10500' GIVEN IMMEDIATE CLIMB TO 11000' AND RIGHT TURN OF 20 DEG TO 30 DEG; DUE TO TWO DEP ACFT AT 12 O'CLOCK CLIMBING TO 10000'. CTLR SAID SHE MADE MISTAKE AND CLRNC WAS FOR THE FLT WITH SIMILAR CALL SIGN BUT SHE HAD USED OUR FLT NUMBER WITH CLRNC; AND WE ACKNOWLEDGED RECEIPT OF IT. NEAREST ACFT (ACR Y) WAS 2 TO 3 NM AHEAD WHEN WE CLIMBED BACK TO 11000' (HAD DESCENDED TO ABOUT 10300') AND WE DID PICK IT UP VISUALLY. CTLR WAS NEARLY OVERLOADED WITH ARRIVAL ACFT AND USING VECTORS AND SPEED CHANGES TO SPACE ACFT FOR APCH. THIS AND TWO SIMILAR FLT NUMBERS CAUSED THIS CONFLICT TO OCCUR. SITUATION WAS CORRECTED PRIOR TO A DANGEROUS CONDITION BUT A FEW MORE SECONDS COULD HAVE BEEN DISASTROUS. REPORT FILED WITH COMPANY TO CHANGE ONE FLT NUMBER SO THIS WILL NOT BE AS LIKELY TO RECUR. WE SUSPECT THE APCH CTLR RADAR CONFLICT ALERT WAS SOUNDED AND A REPORT MAY BE MADE BY THE CTLR INVOLVED. SUPPLEMENTAL INFORMATION FROM ACN #87091: THE APCH CTLR APPEARED TO BE WORKING AT MAXIMUM CAPACITY AND AN ERROR OF ONE DIGIT ON OUR CALL SIGN.