Trigger pull follow through is real important with the Handi, the transfer bar safety mechanism requires the trigger to be pulled fully to the rear when it breaks, or inconsistent pin strikes and consequently, accuracy will be inconsistent. What yours is doing sounds like it may be a loose base, ring or bad scope, so I would make sure all is good from the base screws being loctited and on up. Some base screws have been reported as being too long, or the mount holes have been filled with loctite that prevents the screws from tightening properly, I would check both. The bottom of the mount holes should be shiny, not dull, if they're dull, dig the loctite out and reinstall the bases screws with loctite.
The forend screw torque should be just snug, if you install and "O"ring on the forend stud, the screw should just start to, but not compress the "O"ring. Check out the recent thread on what to do with a new Handi for more tips.
Tim