I know you specified ASA but in Dixie it would be a dead ball out. In the second case, the batter is not out:
:
7:05 THE BATTER SHALL NOT HIT A FAIR BALL WITH THE BAT A SECOND TIME IN FAIR TERRITORY.
EFFECT: 7:05 The ball is dead, the batter is out, and baserunners may not advance.
NOTE: If the batter drops the bat and the ball rolls against the bat in fair territory and, in the umpires judgment, there was no intention to interfere with the course of the ball, the batter is not out and the ball is alive and in play.
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