So it isn't feedback that's the problem. It's a continuous static sound that comes from the amp and stops when you put your fingers to the strings, right?

This definitely indicates a short in the wiring somewhere. Like I said up top, try to isolate the problem. Plug in a different guitar and see if the amp still makes the noise. Try a different cable.

A few more things you could do - unscrew both ends of the cable you are using to connect to the amp. If the soldering or the wiring looks loose, then this could be the problem. It would probably be easier just to get another cable than to repair it. They're cheap enough.

Also, check the plug where the cable plugs into the body of the guitar. There's a little nut around the plug that sometimes will come loose. Make sure it's screwed down tight.

If none of that works, and you've determined the problem to be in the guitar, take it to a music shop or someone who knows about wiring to get it fixed up.