Sorry (maybe it's me; it's late...); how is the p/u connected to the jack plug..? Is it soldered..? Can you post a photo of your connection..? 'Buzzing' will occur if the amp is turned up with no decent connection to its source (the p/u...). The p/u should be soldered to the jack socket, and a screened lead plugged in to the socket, then to the amp. Does it still 'buzz' when connected like that..?
Is there any other way..? I've never thought of it, but I can't imagine how one may not do this (nor yet silence that little voice incessantly rattling away between one's ears...).
I found myself humming a melody, over the space of a few days, and thought, after a while 'What is that..?'. Upon searching a bit, it turned out to be a piece I had, myself, composed a few years earlier..! Dummkopf..!
It's an easy fix for anyone with a modicum of competence. The broken jack plug gets cut off, and a new stereo (TRS...) jack plug soldered onto the cut end of the cable. About five minutes; a suitable jack plug (it's nothing special...) costs a couple of pounds or so for a good'un. Hope this helps.