Mac/Redman,
... But even with factory fodder, 1st shot may be 6" high left, and the 2nd shot may be 8" low right. Shouldn't it form at least some kind of group?
I agree--this is not a group it's a rifle that has serious accuracy problems. I think Fred's on the right trail--check your barrel/breech gap.
Also you ought to try wiggling the barrel on the frame, locked up with the forend off of the gun. The fit of the forend may be masking a sloppy lockup. If there is any up-and-down or side-to-side play, you'll need to return the rifle to the factory for repair.
Of course, if you're the adventurous type you could try Fred's system for accurizing the handi. But I think what you've got is a rifle that has loosened up over time/shooting. The less-than-optimal fit of the barrel in the frame is necessary to allow the frame to accept mass-produced barrels.
Fred's technique of bedding the barrel to the frame eliminates side-play and twisting of the barrel on the hinge; I suspect that your 308's underlug has been peened open a bit and the resulting play is what's causing your shots to go wild.
If this is in fact what's going on, you might be able to shim the underlug in the area it rides the pin and restore the accuracy. If the frame is bedded it will only accept the barrel that is bedded into it, others will need to be hand fitted, but once it has been done your friend's rifle will most likely shoot even better than it did before you sold it to him.
Good luck and let us know how it shakes out.