This is definitely on the right track (as are Metrimate, thanks langwadt!)!
The 24 pin and 12 pin TST connectors seem to be compatible with the connectors on the board, even though it's a different brand. So these will actually work if I can track them down! Thank you so much! The specific numbers for these are ITT CANNON TST12RB06T and TST24RB06T.
Sadly, these connectors seem to be difficult to get your hands on, so if anyone has any clues about what those "compatible off-brands" might be, do chime in. But I can probably figure something out based on this. What a relief!!
EDIT: Alright, barring reputable retailers, I managed to find some on Ebay. Hopefully they'll pan out okay. Thanks again for the help!