by the way ( i just re-read yer post!)
if the problem goes away when you disconnect one pickup, then you have likely stumbled onto the source of the problem!
try swapping the pickup leads around, and then see if disconnecting the same pickup will still stop the noise... if thats the case.. then you might find that the shield connection to that pickup has gone bad. Checking with an ohmeter is possible, to see if the shield connection is flakey.
I have an AXY on my desk here, and on the bottom, where its encased in epoxy, i can see four tiny spots near the edges of the pickup bottom where the copper faraday shielding protudes out of the epoxy a little... (i can see 4 minute pinpoints of exposed copper) If you can find them too (or just carefully scrape a little epoxy off to expose them) then you can at least test for continuity with the pickup leads.
The AXY in front of me has some copper shielding exposed just where the lead enters the epoxy, and actually, on this one it looks almost like the shield part of the wire lead isnt connected there by very much! (hanging by a thread, even) I could see this being an area where a problem could occur...
also, while you have the ohmmeter handy, compare the resistance reading of the good pickup with the resistance reading of the noisy pickup.. if its different by more than 5% - that could indicate a problem with its wiring lead (or even its internal wiring, though thats very unlikely)
If you isolate the problem to one pickup, then you are halfway fixed! IF the shield looks good, i would try ferrite beads on that pickup lead. If that fails, probably ordering a new pickup will be your simplest solution. (assuming the other pickup is noise-free when connected to either position )
hope this helps a bit :-)