If the team drops to 7 on defense, and the DP is still in the game (or has a re-entry available), they may continue by having the DP play defense for one of the absent players. This gives them 8 on the field.
They cannot continue, under any circumstances, with only 7 on either offense or defense.
While the shorthanded rule has not caught up with the DP/Flex addition, clearly the rule is written for the possiblity of only one vacant position in the batting order.
Quoting from the 2003 book (assuming it is the same in 2004 - I don't have my '04 book yet)...
NFHS Rule 4-3 ART.1 ...
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g. A team shall be allowed to finish a game with 8 players (F.P.)... with an out being called in that spot of the batting order. If that player has safely reached base ...
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I would not allow a game to continue with only 7 in the field or with only 7 in the order.
If the coach wants to protest, fine.