Hey guys, thanks for the responses. To clarify, Sanderxpander is correct - I mentioned multiple cards purely for testing purposes. I am only using one at a time, and I'm not using ASIO4ALL. I'm using ASIO with all devices but with the manufacturer's drivers.
I've also found that this problem isn't isolated to Melodyne - it happens with particular plugins. I've discovered that Slate Digital Virtual Tape Machines is also affected by the problem, which kinda sucks because it sits on the master track and has children on all tracks. I can always bounce the Melodyne Region FXs to clips (as suggested, and I generally do, but I'm trying to troubleshoot this thing), but the VTM is a more annoying and permanent problem.
So it seems like certain plugins are causing some kind of feedback. I can't find any literature on this anywhere, and even figuring out the correct search terms for something like this is difficult. Going to update the thread title and original post with new information, and clear up the confusion with sound card(s).
EDIT: additional info - if I switch the RME Babyface from ASIO over to WDM, the problem persists. So it's not a driver mode issue. Also, I've just discovered that the issue DOES occur with my FireStudio Project, but it's way less prevalent - to the point where it's hardly noticeable unless you know what to listen for. But it's actually good to know that it's happening across all devices, in a twisted way...