I did a bit more on the diagnostic end. I found that there appear to be two distinct causes of buzz that I am getting here.
The first is a grounding issue of some kind, and is mostly prevalent in the neck pickup - although I can (barely) hear it to a much lesser degree with the bridge pickup. I noticed that when my hand is either touching the neck pickup or hovering above it, I get a *lot* of buzz. The greater the surface area of my hand involved, the stronger the buzz.
This is alleviated when my other hand touches the bridge or strings - thus it seems to be a grounding issue.
Experimenting, I opened the electronics cavity and found these electronics inside
There is actually only a single white cable that runs out of the pickup, that splits off into the ground and signal wires right before they are soldered to their posts. While these connections look solid on the electronics end, I found that touching a screwdriver I was holding to the ground post that the neck pickup was soldered to stopped the hum immediately.
This leads me to believe that either the ground wire inside the white cable is degraded or it is degraded where it connects inside the pickup itself. Regardless, I'm not sure how to fix this issue as when I pull out the pickup it seems to be one solid block with the white cable embedded into it - there are no screws or anything to remove the cover... maybe there are under the foam that's glued to the back?
I tried lining the cavity with aluminum foil to a degree and running a piece of aluminum foil under the pickups and under the bridge (in a way that looked aesthetically pleasing as it creates a stripe across the center of the bass) and while this seems to have helped minimally (and actually enhanced the tone of the neck pickup, strangely), it hasn't fully alleviated the issue.
What *does* alleviate the issue fully is when I run my bass amp from battery as opposed to off the wall socket (it's a roland microcube, capable of either). Then there is no hum at all caused by my body touching or hovering near anything.
The second issue seems to be different. Whenever I turn on the lights in my living room, I get a similar but much stronger buzz that is equal in both pickups (although its tone changes subtly when I roll from one to the other).
The lights are on a dimmer switch, and turning them down makes the buzz much worse than when they are on full. The only thing that stops it completely is turning the lights off.
This buzz cannot be alleviated by touching anything with my body, and cannot be alleviated by running off of battery. Changing the direction that the bass is facing helps, but does not fully alleviate it -- again, the only thing that can get rid of it is turning the lights off.
So... any advice on how to proceed?