Don't worry about the paper not slipping down the channel YET. Your gun may very well shoot nice groups without "fixing". I would have the trigger worked on to get it to the 3 pound range, crispness is more important than overall poundage.
First mechanically center the scope.
Turn windage turret all the way to left, never force it, not even one more click than it wants to go. Then turn it all the way right while counting clicks. Turn back to left 1/2 the number of clicks.
Repeat for the elevation turret, all the way up, all the way down while counting, 1/2 back up.
Make sure mounts and rings are tight and that there is no obvious binding between scope, rings, mounts.
Bore sight -
Remove bolt
Rest rifle on solid rest
Align center of bore with point of reference on target
Don't touch!
Verify if crosshairs match point of aim for the bore
Do this at 100 yds or so if possible
Do this several times
THEN adjust crosshairs to meet point of aim for the bore
Replace bolt
Shoot 3 rounds at 25
If close, it usually is...
Move to 100 wait for rifle to cool
Shoot three more
Adjust, wait to cool
Three more, adjust if needed
Alternatively,
Mechanically center
Fire one at 50
Move crosshairs to match point of impact
Shoot three more
Move crosshairs to center of group
Fire three at 100, adjust if necessary
Biggest issues is stability, comfort and repeatability.
You need really stable rest for the front end. Don't let the stock touch the table. Use another bag back there.
Finger tip squeeze on the trigger. Consistent check weld on stock.
If you can't get workable groups, I'll send my address. Just send the rifle to me...
ngh