GND CONFLICT BTWN 2 TAXIING B737 DURING HIGH ARPT ACTIVITY AT DEN; CO.

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PUSH OFF GATE 2 MINS EARLY ON A LINE CHK. KNEW THAT WE WERE LIKELY TO GET THE SHORT TAXI TO RWY 17R; SO WE WERE SET UP FOR THAT RWY. NO COUNT FROM THE FLT ATTENDANTS; SO WE HAD TO CALL THEM. GOT THE COUNT AND THE NUMBERS IN AS WE APCHED SPOT X; BUT LOST MORE TIME AS THE LINE CHK CAPT STARTED DISCUSSING THE CHANGE TO THE FORMER REQUIREMENT THAT WE ENTER MAC FROM THE ACARS TKOF NUMBERS. CHKED IN WITH A VERY BUSY GND CTLR AND TOLD 'TAXI STRAIGHT AHEAD FOR RWY 17R AND SWITCH TWR.' READ THAT BACK AND TOLD HIM THAT WE WOULD NEED A MIN. HE REPLIED 'ROGER.' TOLD THE CAPT I WAS GOING HEADS DOWN AND DIALED UP TWR. WAS FEELING LOTS OF TIME PRESSURE AS WE STILL HAD TKOF CHKLIST TO DO AND WE WERE NEAR THE HOLD SHORT. LINE CHK CAPT LATER SAID THAT GND SWITCHED US TO RWY 17L JUST BEFORE WE SWITCHED OFF FREQ. NEITHER THE CAPT NOR MYSELF HEARD THAT AND WE NEVER RESPONDED TO THAT CALL. TWR SAID TO SWITCH BACK TO GND. WE DID AND THEY GAVE US TAXI TO RWY 17L INSTRUCTIONS THEN SAID 'STOP.' WE DID SO SLIGHTLY INTO THE CS AND TXWY M INTXN AND WATCHED WITH SOME CONCERN AS A B737 TAXIED IN FRONT OF US ON TXWY M. GND WAS CALLING HIM AND GOT A RESPONSE AFTER HE PASSED US. SUPPLEMENTAL INFO FROM ACN 594755: GND CTL (121.85) SWITCHED US TO TWR FREQ 133.3. RIGHT BEFORE SWITCHING GND ISSUED A TAXI CLRNC TO RWY 17L VIA TXWY M AND ED; BUT DID NOT STATE A CALL SIGN. WHILE ON 133.3; TWR DIRECTED US TO RETURN TO GND FREQ. AT THIS TIME WE WERE ON CS JUST ENCROACHING ON TXWY M. TO OUR R TAXIING NBOUND ON TXWY M WAS AN ACR FLT. THEY APPEARED TO BE TAXIING AT A FAST PACE. GND DIRECTED THEM TO GIVE WAY TO OUR ACFT AND DIRECTED US TO PROCEED TO RWY 17L VIA TXWY M AND ED. THEY DID NOT SLOW AND PROCEEDED TO PASS US EVEN THOUGH THE NOSE OF OUR ACFT WAS CLRLY ON TXWY M. THE WINGTIP PASSED WITHIN APPROX 7 FT OF OUR ACFT. THE NOSEWHEEL OF THAT FLT WAS CLRLY AND WELL OFF TO THE R OF THE TXWY CTRLINE. WE WAITED FOR THEM TO PASS THEN PROCEEDED TO RWY 17L WITHOUT INCIDENT. GND CTL TOLD THEM THAT HE HAD CALLED THEM 3 TIMES TO GIVE WAY. THEY RESPONDED THAT HIS XMISSIONS HAD BEEN BLOCKED.