You can't just switch leads on pickups with metal covers. The covers are connected to the ground wire, so when you try to use that as the hot wire, you get all the interference they're picking up mixed in with your signal.
You need to disconnect the pickup cover from the wire it's connected to and attach it to the other one. It can be done, but it's delicate work. Do some searching and you'll find a bazillion threads about this, including ones with much better instructions than I can give you. Good luck!
Originally Posted by jondanger
My first thought is to switch the leads on BOTH pups this time. Is that the right move? Will that just make the neck pup cover hum?
No and yes.