A B737-700 CREW FAILED TO COMPLETE BOTH THE SHUTDOWN AND BEFORE START CHKLISTS; LEAVING THE HYD PUMPS ON; AND SUBSEQUENTLY BROKE 2 TOW BARS.

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ABOUT 30 FT INTO THE PUSHBACK FROM GATE; WE FELT THE TOW BAR SNAP. THE TUG OPERATOR CONFIRMED THAT ON THE INTERCOM. THE GND CREW QUICKLY HAD ANOTHER ONE HOOKED UP AND WE RECOMMENCED THE PUSHBACK. #2 HAD BEEN STARTED DURING THE FIRST ATTEMPT; AND #1 WAS STARTED DURING THE SECOND. AFTER ABOUT ANOTHER 10 FT; THE SECOND BAR SNAPPED. GND CREW CONFIRMED THIS AND RPTED THE SECOND BAR CLR AND NO ACFT DAMAGE. I ASKED THE RAMPER TO STAY ON THE LINE WHILE WE SORTED THINGS OUT ON OUR END. DURING THE AFTER START FLOW; WE DISCOVERED THAT ALL 4 HYD PUMPS WERE ON. WE HAD SUFFICIENT CLRNC TO TURN OUT SINCE THE ADJACENT GATE WAS UNOCCUPIED. AFTER ASKING THE GND CREW TO WING-WALK THROUGH THE TURN; I CLRED THEM OFF AND TAXIED OUT AFTER RECEIVING THEIR THUMBS-UP CLRNC. WE WERE LATE GETTING THE BEFORE START CHKLIST DONE BECAUSE WE WERE ATTEMPTING TO DETERMINE WHAT THE RVR WAS. I HAD ALREADY GOTTEN A REVISION TO MY RELEASE FOR A TKOF ALTERNATE. WE HAD BEEN SCHEDULED FOR OVER 1 HR ON THE GND AND THEN GOT IN 35 MINS EARLY FROM MCI. WITH ALL THAT GND TIME; WE REQUESTED GND PWR; WHICH TOOK A COUPLE OF ATTEMPTS TO GET HOOKED UP. IN THAT DISRUPTION OF THE NORMAL ROUTINE DURING PARKING; THE ENTIRE PARKING CHKLIST WAS PROBABLY NOT ACCOMPLISHED AND THE HYD PUMPS HAD BEEN ON THE ENTIRE TIME. IN RETROSPECT; I REALIZE THAT WE WERE IN A SITUATION OF NEITHER GATE SVCS NOR STRAIGHT APU PWR; AND GETTING IT SORTED OUT DISRUPTED/DISTR US ENOUGH TO NOT COMPLETE THAT CHKLIST. SAME THING HAPPENED ON THE PUSHBACK EVOLUTION WITH THE WX DISTR.