Overtones or Feedback

I notice on acoustic guitars one can typically hear a high pitched metallic ringing sound when a string is picked and quickly damped. Sort of a "clank." It comes from the section of string between the nut and tuner post, and goes away when that string section is damped. Perhaps you're hearing a similar ringing, and the pickup is sensitive to that frequency because it's near its resonance? (I have an acoustic with a piezo pickup that I can't play with slide because the slide makes a high squeaking noise on the string, normally inaudible, that resonates with the pickup to make a horrible squeal.) Try sticking some foam under that string section above the nut to damp it and see if it goes away.

I've got an early 90's Fender Jazz Plus with active Lace Sensor pickups. My problem is that I get a fairly annoying high pitch on my G string after I fret a note. Basically, the fundamental goes away after a few seconds and this high pitch sustains. It does not happen on the other strings.

High pitch like a harmonic? I had that on a bass A string. It wasn't seated properly in the nut. Or maybe it was the bridge saddle .... don't remember. Do you only hear it amplified, or is it also there if you just play un-amped?