Is it possible that there's something not quite right here, some ground-loop or broken-connection short issue?
I took the pickguard off & it looks solid in there. There's a tiny square of foil on the back of the pickguard under the pots, but no other shielding at all.
Are the metal-covers connected to ground?
There is continuity between the metal cover & ground, though it's not a dead short, more like 12 - 14 Ohms. Touching a wire from the pickup cover to ground made no noise difference just now.
These Danelectro-"lipstick-tube" style pickups usually aren't THAT bad... ? Usually, their metal covers- grounded- actually provide better shielding than many single-coil pickups get...
This is bad, like the recording I attempted today I can't use at all, even after running an extremely expensive, powerful noise reduction software on the track. And finding the proper positional polarity in the room.
I did that process for The Residents' Gibson Melody Maker before we went out on tour many years ago. It was a Starz Guitarz kit. The improvement was very dramatic. That's certainly a good place to start.