I'd be suspect to the scope/mounts as well. I use nail polish on all my screws when mounting scopes. Problem followed both barrels, so I'd suspect something else. Or is the scope being taken from one barrel and moved to the other?

I know on some of the T/C Encore's the hinge pin was/is too small and the guns don't lock up right. There are several different companies that sell oversize pins to help correct this problem. Don't think the H&R has the same design, but wonder if something in the forend or how it is attached to the action is lose or possibly not in proper head spacing for the cartridges.