diagnose first, replace after you know
i won't disagree with the suggestion to change to this or that... It could be a valve. You said this only happens when water moves.
a,) "knob style shut off" --- that is a valve
b,) it could be the toilet water tank inlet valve.
c,) or even a pressure reducing valve.
All are valves, and all can be happily replaced with the newer and better alternatives suggested. Turning off valves and "seeing what happens" is a scientific process of methodical trial and error, assuming you keep track of the permutations and combinations. This will help you isolate the most probable causes, and eliminate some of the possibilities too.
Post what you find out.