It sounds like it might not be the packing. Usually, you can get two wrenches and hole the faucet with one and turn the large bolt just behind the handle and it will crack down alittle. It it is not moving, do not mess with it, you could bust the thing off the wall.
I would, yes, turn off the water main, then remove the guts and take it in and have a plumbing store associate take a look see and determine the problem.
Like pb said, it's now your packing or washer or seat if one.