ground hum driving me insane. really.

Oh you guys will have probably heard this a dozen million times, but I need help with this ground hum. I rilly do.
Story: I wanted swap the body on my Baja Tele for something lighter and decided, while I'm at it, to uh...simplify that S1 switching. I love the Baja neck (as in really love it) and the pickups, so a change of wiring and body made sense.
Got a very nice body from Axesrus, and some pots and a cap etc.
Used the Seymour Duncan Tele '66 wiring diagram.
Soldered it all together carefully.
Yikes: loud ground hum.
Tested with a multimeter for continuity from various points on the guitar to the jack shield on a short lead - neck pup cover, bridge, knobs, all grounding solder points inside, strings, you name it - still the loud noise.
It stops completely when I touch the strings or any other metal part with the chunky silver ring on my middle finger.Yet those same metal parts show continuity to ground on the multimeter.
I know little about matters electric, but I don't understand how this can be?
The guitar sounds fabulous incidentally - when there's metal touching metal :roll:
Some help would be really welcome, thanks.

That is supposed to happen to some extent, but if there's a very noticeable difference in hum I would check every solder joint to make sure everything is star grounded correctly.