thorr1965:
The problem is, either the trigger assembly is set too deeply into the stock (very likely) or the screw that some locks have to adjust the tension on the rear trigger spring needs adjustment (not all trigger assemblies have this). This is causing the rear trigger blade to sit too high when the rear trigger is not cocked.
If the adjustment screw exists, it is the first thing that should be adjusted. To see if the trigger assembly has one, look back behind the rear trigger. If a screw exists there, screw it in (clockwise). If the screw doesn't exist, read on.
Very often on older guns the trigger assembly will have been crushed deeper and deeper into the mortise by people overtightening the tang screw.
Check out the area at the rear of the mortise where the rear of the trigger rests. If it is the flat surface I think it is you will need to make a shim to put between the flat area of the wood and the underside of the trigger plate.
I'm just guessing but it will probably need to be about 1/16" thick. (Several thicknesses of cardboard like a cracker or cereal box will be needed. A thin piece of wood is even better).
Once placed between the trigger and the stock wood, screw the trigger assembly back in place. You should now be able to cock the gun without needing to set the rear trigger. If you can't, try adding a few more shims.
A more permanent way of fixing the problem is to file the top surface of the rear triggers blade down. This will lower the upper surface of the trigger blade with respect to the sear arm on the lock which will allow the sear in the lock to engage the full cock notch. (This is commonly done when one is installing a new set trigger in a new build of a rifle. It is just part of the game.)
As for the screw between the triggers, it does NOT control the location of the rear trigger blade with respect to the locks sear arm. Adjusting this screw only controls the engagement of the front trigger with the rear trigger to hold it in a "set" position. It should always be adjusted to assure a full engagement 100% of the time when the rear trigger is pulled back against the spring pressure.
To do this adjustment, screw the screw out at least 4 full turns. Then, set the rear trigger. Now, screw the adjustment screw in until the rear trigger is suddenly released. Once this happens, back the adjustment screw out 1 full turn. (Lyman and a few other companies provide the trigger assembly with a screw that is too short to actually release the rear trigger. They seem to think this is a safety measure to keep people from incorrectly adjusting the screw so, if you can't get the screw to release the rear trigger just screw it in so it doesn't get lost and call it a day.)
Try these fixes and let us know what happens.