ACR HAS RWY CHANGE ON TAXI AT MIA WHILE TRYING TO GET WT AND BAL INFO. TURNS TOWARD WRONG RWY; GND CTLR CATCHES ERROR.

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WE WERE CLRED TO TAXI OUT TO RWY 9L (NO SPECIFIC TAXI RTE GIVEN). DURING TAXI OUT OUR OPS WAS SLOW PROVIDING THE WT AND BAL NUMBERS FOR OUR FLT OVER THE ACARS. THE FO CALLED TO OPS WHILE (APCHING THE RWY) TO OBTAIN THE NUMBERS FOR TKOF WHILE I REMAINED WITH GND CTL. DURING THIS TIME I WAS MENTALLY CALCULATING THE TKOF WT FOR A CHK WITH THE OPS NUMBERS AS I HAVE PREVIOUSLY RECEIVED INCORRECT NUMBERS AT THIS STATION. IN THE MEANTIME GND CHANGED OUR TKOF RWY TO 12 AND SWITCHED US TO TWR WHICH I ACKNOWLEDGED AND SWITCHED (FO STILL ON OPS FREQ). THE FO GOT THE NUMBERS AND THE CHKLISTS COMPLETED AS WE WERE CLRED FOR TKOF. (STILL A WAY OUT FROM THE RWYS BUT RAPIDLY APCHING THEM.) I QUESTIONED THE GROSS TKOF WT AND ASKED THE FO TO RADIO OPS AND CONFIRM IT; WHICH HE DID. THE RWYS AT MIA ARE THUS. WHILE WE WERE CLRED FOR TKOF ON 12 I STILL HAD RWY 9L IN MIND FROM THE INITIAL CLRNC AND ACTUALLY BEGAN TO PULL ONTO 9L WHEN TWR CALLED US AND SAID TO MAKE A L AS THAT WAS 9L AND WE WERE CLRED FOR 12 (THERE WAS AN ACFT ON ABOUT A 3 MI FINAL FOR 9L). I THEN MADE THE L AND MADE THE TKOF ON 12 BUT DID ENTER THE RWY 9L SLIGHTLY BEFORE BEING CALLED BY TWR. ALL THIS TOOK PLACE IN LESS TIME THAN IT TAKES TO TELL. ONE OBVIOUS PROBLEM WAS THE WT NUMBERS DISTR. ANOTHER WAS THE PHYSICAL LAYOUT OF THE RWY AT MIA AND THE USUAL LIMITED MARKINGS IDENTING THE RWYS. OF COURSE BEING CLRED FOR BOTH RWYS DID NOT HELP. SOLUTION: STOP DISTRS; OR STOP UNTIL THEY ARE CLRED UP -- 2 BETTER RWY MARKINGS.