Constant buzzing but with sound and the controls all functioning normally is due to having the signal and ground wires swapped at the output jack, though it can be down to a bad ground connection somewhere in the wiring harness.

If you've got a multimeter, check if there's continuity between the metal nut on the outside of the output jack and the back of a pot. If not, it's likely the ground and signal connections are swapped on the jack. Easy to do.