B737 CREW CATCH A MISTAKE IN A CTLR ASSIGNED ALT JUST IN TIME TO PREVENT A SERIOUS CONFLICT IN ZFW AIRSPACE.

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WE WERE ON THE WORTH 3 DEP OUT OF DAL (DALLAS LOVE); ABI TRANSITION. ABOUT 20 MI E OF ABI; WE WERE GIVEN DIRECT MAF. WE WERE AT FL200; AND WERE GIVEN A CLRNC TO CLB TO FL280. I READ BACK 'FL280;' DIALED IT IN TO THE ALT SELECTOR; AND THE CAPT REPEATED IT OUT LOUD; '280.' WE WERE NOT IN A VERY QUICK (VERT SPD) CLB; WHICH TURNED OUT TO BE GOOD. AROUND 26800 FT; TCASII WARNED US OF THE TFC UP AHEAD. (WE WERE ALREADY WATCHING THAT TFC). I SAID OUT LOUD; 'HE SURE SEEMS TO BE CLOSE!' WE DID A DOUBLE TAKE AT THE TCASII MONITOR; AND THAT PLANE WAS 1000 FT ABOVE US (GOING R TO L); WHICH WAS OUR ASSIGNED ALT! (FT WORTH NEVER SAID ANYTHING OF THE TFC TO US.) CAPT QUICKLY SAID; 'VERIFY OUR ALT;' AND CHOPPED THE PWR; AND LOWERED THE NOSE. I ASKED ZFW TO 'VERIFY ASSIGNED ALT' FOR OUR FLT. CTR SAID 'MAINTAIN FL270!' (CAPT TOLD ME; THAT IS NOT WHAT THEY GAVE US; THEY TOLD US FL280!) I THINK AT THAT TIME; CTR POINTED OUT THE TFC (NOT SURE ON THAT). WE STOPPED THE VERT CLB AT 27200 FT; AND EASED IT BACK DOWN TO 27000 FT. ATC SAID NOTHING TO US. THE OTHER PLANE SAID NOTHING. AFTER THAT TFC WENT BY; WE WERE CLRED TO CLB TO 35000 FT. THE REST OF THE FLT WENT NORMALLY. I THINK THE GOOD CRM AND COM BTWN CAPT AND I; HELPED SAVE THE DAY. AND LET'S HEAR IT FOR TCASII; WHICH GAVE US AN ALT READOUT OF THE OTHER GUY. I AM NOT SURE WHY CTR DROPPED THE BALL.