Mt Al,
Do you still have the double rifle? If so, instead of waiting between rifle shots (at targets) you should shoot the second shot as soon as you can get back on target. Don't wait to see where the first bullet went if you are checking if the barrels are crossing or shooting "wide". You can check this without slowing down, by using two targets next to each other. Shoot the right target with the right barrel and before the sound goes away shoot the left target with the left barrel. Then let the barrels cool (the cold shot barrel will affect the warming rifle barrels otherwise). After you have shot three rounds from each barrel (cold) put one target on top of the other and mark the holes in one onto the other. What each barrel is doing will them be clear.
If the front base itself is moving, it may need to be re-soldered, but if the plate is moving in the dovetail, it can be tightened up a couple different ways. If the scope itself is moving, correction depends on the type of rings (soldered half ring, screwed full ring, or dovetail). BTY, I understand your discomfort with set triggers, but experience will remove that. Just set the trigger when you are ready to shoot. My German gunsmith friend cautioned that it is more dangerous to hunt with the safety on and the trigger set than with the safety off and the trigger not set. If you set the trigger and don't shoot, open the gun and unset the trigger. Some people unset it without opening the gun but a "slip", results in a shot. If it is open the firing pin can't reach the primer. I use set triggers on most of my guns, by just touching them when I am ready to shoot. If you are used to squeezing the trigger set triggers are entirely different but are helpful (but not for the second shot with a double rifle). You said you try to keep shots with a drilling under 100 yards. That is fine, but I find that drillings are surprisingly accurate (for one shot) because the rifle barrel being bound to the other barrels makes it very stiff, in spite of being very small in diameter and very light.
Good luck,
Mike