I doubt fret noise comes through in your line playing. Probably when you change position?
I had such problem everytime when it was the time for me in the song to stop playing completely, ie a pause, the short period between changing board position or fretting a new chord or the even shorter period between releasing one note/ fret and playing the next in a line passage.
What I do now is dampen the strings with both hands and hit mute on the switch when I'm not supposed to be playing. When I'm playing I try and dampen with my picking hand like people do during fast picking shred passages, aka muting.
A way to tackle this is practice with an acoustic or with the electric in high gain to spot the problem and then try to eliminate it. the way for me was using ch.3 and be cautious about right hand muting and how I release my fingers after fretting a string.
I had to do that after I listened to a recording of mine and realised how noisy I was when I was not playing. This was sympathetic noise when I was changing positions or when I was leading in towards a pause. Although at this stage I was a decent player, I had never spent much time practicing using dirt, or using an amp even, and of course practicing at home means you can't crank the amp to volumes which allow every tiny noise to become noticeable.
so I'd guess you somehow need to recreate the conditions which will allow such problems to come through in practice-friendly volumes and then tackle them.
