So I took The Shadow's advice (thanks!) and ripped the audio from my original Blu-ray and my replacement Blu-ray, and imported the resulting DTS files into Audacity. Interestingly, it looks like the old disc not only had an empty channel for the left rear speaker, but another channel as well (I'm guessing the subwoofer/LFE channel). That may have been previously reported, but that I only remembered that the left rear channel was blank. The good news is that it looks like the replacement disc does in fact correct the audio issues that the old discs had. I've included links to images of the audio tracks in case you'd like to see them. Also, I'm no A/V expert, but from my reading online it looks like DTS-HD has two audio tracks so that BD players can still be compatible with old receivers, which is why there are two DTS files from each Blu-ray. I have included screenshots from both DTS files from each disc, even though the audio files for each disc look identical (granted, this could have been an issue with the software I used to rip the audio).
I also noticed a difference in the packaging between my new disc and old disc. My replacement disc has the "Dolby Digital" and "dts-HD Master Audio" logos located on the back cover, to the right of the rating and above the Blu-ray Disc logo. The release date disc lacks these logos. Its possible that this could indicate which disc you have, but I can't be certain of that.

EDIT: After taking my photo and editing it to point out the differences, I noticed another change between the covers as well. On the release date disc, on the back cover under the film information (MIRAMAX PRESENTS A TOHO PRODUCTION...), there is copyright information that says "© 1992 MIRAMAX -
http://WWW.MIRAMAX.COM". You can see that line from my original cover in the photo that I posted. My replacement disc instead says "© 2012 MIRAMAX -
http://WWW.MIRAMAX.COM". I may post a revised picture later showing both differences in the covers.
tl;dr: My replacement disc from EchoBridge does have corrected audio. Contact them to get your replacement now!