A 1969 vintage, in my opinion, suggests the probability of necessary and thorough cleaning before thoughts of tired or worn out main springs would be the cause of the malfunction you have detailed. The shotgun you speak of is fully cocked when its operator closes the breach. It is a single trigger mechanism with an actuator switch to select which barrel (upper or lower) fires first. Quite often, especially if the actuator switch mechanism is dirty or out of sync because it is dirty, one operator or another may experience the same sort of problem.
Remove the butt to expose the trigger group. Use a vacuum cleaner with hose to remove any loose debris from the machined wood of the butt and any debris (wood chips etc.) that may have taken residence inside the shotgun’s trigger group area. Clean all metal with a soft bristle brush (tooth brush, small paint brush, shaving lather brush, etc.). Wipe off all grease and crud wherever with Q-tips, pipe cleaners or toothpicks. Sparingly apply your favorite low viscosity oil used for things like sewing machines and hair clippers to pivot points and contact faces –no penetrating oils or dry lubes- then wipe away any excess with Q-tips. A light film is good everywhere but dripping anywhere is no good. Load snap caps in each chamber and close the breach. Set the actuator to fire the barrel of your choice. Squeeze the trigger, being careful to avoid touching any of the moving parts inside the now visible mechanism and watch what happens inside. Inertia, as a result of recoil after firing one barrel or the other does not cock the shotgun but it does set the trigger for the remaining barrel even in a dry-fire exercise, i.e., if all the clockwork inside is clean. If the trigger actuator is out of sync, you can remedy that too by selecting a different barrel after opening and closing the action again, then pulling the trigger twice to make it happen. Watch what the trigger actuator does when you move it. What else moves when you operate the actuator switch? Look at and learn the inner works of your shotgun. But never venture beyond the realm of your own mechanical aptitude or the availability of proper tools for the job. It isn’t a Swiss watch but if you treat it as such you sure won’t hurt anything.
On the butt -the parts on the inside that were machine cut- lightly sand with 320-400 grit paper any sharp corners and furry looking areas. Then, copiously apply fine Furniture oil to any parts of the inside of the butt that didn’t receive the same treatment as the wood on the outside and allow it to soak in.
Re-assemble the shotgun when you’ve decided it’s time, or can’t wait any longer.