Alright, so I downloaded a CC default replacement skin that apparently messes with the baby's appearance (didn't realize this until after I had gone through all my CC to find the "bad" CC). I deleted the file, the cache and all of that, and went to play the game.

Newly made sims have no issues now, however, I do have a save file from when I used this CC and both parents had the CC skin, and it seems to have transfered to the baby despite getting rid of the CC.

Before I removed the skin, the glitch (the baby had no eyes) would be present in "Edit in CAS" (Nraas Master Controller). Now, the baby looks normal in CAS but has no eyes when I go into live mode. So I assume that the skin has been removed from the game but, somehow, not from these sims/this save file. Is there any save related cache that I need to remove? I don't have a Saved Sims folder, so I can't do that; do I have to go into the DCCache folder? Or Saves (where I extracted the save file from the backup)?

Offhand, I would say what you really have is a corrupt sim there. I would go about it the way you were describing in a different thread here, by re-creating the baby such as by using MasterController's Instant Baby function and TAing (Totally Annihilate, it arranges it so it's as if the sim never existed) the first one. Is that idea not working out?