I am sure that in a case of a increased hum its a bad connection to ground. Resolder the connection and check the path to ground. Check the connection from bridge/trem claw to ground is good. About shielding: There are different kinds of noises which can get into your system like radio frequencies, 60 Hz hum, static noises and so on. First: you can't eliminate them completely even if you use active pickups with a high impedance - you can reduce them more or less. With copper foil you can eliminate the noise and the RF quite much and reduce the hum a bit. But its a matter of doing it right: Use copper foil with conductive glue, make sure that all points of the laidout copperfoil to the jack ground are good, twist the pickup wires and use a shielded wire between the volume pot and the output jack.