I prefer the second version
Of the two, the second construction sounds more natural and immediate to me than the first ("that" impedes the flow, and in general should be avoided if at all possible). For the same reason the dash should be omitted, as another poster has commented, but I believe a dash is needed between "freshly" and "opened." You might also consider a more direct rewording that conveys the same thought:
"As if it were dying, the voice in her brain drained [ebbed?] in strength like the dark blood leaking [oozing?] from her freshly-opened leg wound."
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