If it stops when you touch the strings, then the bridge IS grounded. If the bridge was not grounded, it would not stop.
Some amount of hum is normal with any guitar with passive electronics. It will be worse with single coil pickups than with humbuckers, obviously.
Shielding the cavities may help, in fact, it will probably help. It is not a cure all, however. Give it a try, and if the problem is better, then great. If it still needs work, look at the electronics (in fact, do this first). Do all of the connections look good? If so, then it becomes had to figure out the problem. I would try shielding it, and if it is still a problem, you need to talk to a good repair person. They should be able to figure out the problem.
Light
"Cowards can never be moral."
M.K. Gandhi