I have 120v when measuring between true ground and the screw on my outlet plate
Unfortunately there could be several causes.   At a minimum you don't have an effective grounding conductor, to say the least.  That's one thing that needs to be fixed.  A way to find the problem is to unplug everything on that circuit and see if you still have the problem.  If not, you have a defective (ground faulting) appliance or other piece of equipment. If you still have 120 between ground and the real ground (the Earth!), then you have a short someplace.  I'd try removing all the outlets and examine them for shorts, loose wires. 

I recently fixed a true short (tripped the breaker when a outlet controlled by a switch was turned on). Turned out over time the ground wire in the box came into contact with the hot side of the outlet.  

If you are comfortable working in a main panel, I'd pull the cover off, determine which breaker controls the "shocking" circuit and follow the wire from the breaker to where the cable enters the panel, then follow it's ground and see if it's connected and in good shape. 

You could also have a break in the ground connections anywhere from the main panel throughout the circuit. Like Ed, ph3, Harp and others have said here, wire and cable failures inside a wall, once installed are extremely rare.   By far most often these issues are in an outlet or fixture box someplace.   One more reason the code requires fixture and outlet boxes to be permanently accessible!   

Please be careful when working inside a live panel.  One of my rules personally is only one hand in the panel at a time whenever possible.