Would shutting down the engine have prevented the Atlantic Southeast Airlines Flight 2311 crash?
The description of the accident says the propeller went to a flat pitch of 3°, which is still positive. At such angle, the air is trying to spin the propeller in the same direction as the engine, so the engine running helps the propeller spin. Only if the propeller went all the way to negative pitch would the engine be the air trying to spin it in opposite direction and therefore produce additional drag, but it appears that propeller was not built to allow negative pitch.

So I don't think it would have helped. It probably wouldn't have made much difference in the very flat pitch.