Hi Sydney,
Assuming that your At-home GMAT Score is one that you are happy with (and you submit that Score to the Schools that you plan to apply to), the only way that those Schools will have proof of your AWA6 from your prior attempt is if you ALSO submit that prior result to the Schools. In general, Business Schools don't care if you take the GMAT more than once, so having more than one Official Score on your record shouldn't be an issue. With a 2nd Score that is significantly higher, you can actually 'spin' those 2 Scores as a 'positive' (re: you don't give up, you overcome challenges and improve, etc.).
All things considered, most Schools don't put too much emphasis on an Applicant's AWA Score (as long as you score a 4.0 or higher, then that would be acceptable), so that AWA6 won't radically improve the strength of your Overall application. In addition, since the current version of the At-home GMAT has no AWA component, Business Schools have almost certainly adjusted how they will view your results from an At-home attempt to account for that missing piece of information (and by extension, you don't have to worry about it).
GMAT assassins aren't born, they're made,
Rich