So after a fair bit of frantic troubleshooting to find out what was draining all of my FE, I managed to trace it down to the culture vats and analyzers from Fossils. Apparently they consume power even when they are empty/not being used, and they don't appear to be able to be turned off via redstone signal or anything else.

Not sure if this is an intended behavior, a bug, or an oversight. There are simple enough work-arounds for it depending on what mods are being used to supply power, but it would be nice to at least be able to turn the machines off directly when they are not in use.