may be you can try shooting a sheet of plain white paper using AWB, without changing the lighting or the position of the camera, then shooting continuously with or without flash.
for may be 10-20 shots, check if the AWB is consistent or not...
if the WB still varies by a lot, then just too bad.....listen to ESPN recommendation - shoot NEF or use custom WB, then use Nikon capture to tweak the color or just change the camera to avoid all those troubles