Lock operation:
There is a vacuum tank in the LEFT FRONT fender (behind the rocker panel) that stores vacuum for when the car is not running.

The doors should ALL lock *AND* unlock when the driver's door is. This is the same with the gas door AND the trunk as well. The gas door requires a vacuum to stay locked, so if your system leaks, it will not stay locked.

The driver's door lock CANNOT be pushed down when the door is opened. Idiot-proofing, German style. You need to use the key, or open the rear passenger door and lock that way.

If your locks only work when the engine is running, and don't even work properly when it's shut off you have a MAJOR vacuum leak. That's normal. The 40 year old rubber diaphragms don't hold up. You can find replacements, or clean yours really well and coat them with a layer of silicone. The slow lock is probably a bad diaphragm. The door locks are spring-loaded so they can be locked/unlocked, with resistance, without moving the diaphragms, so this sounds like what you're feeling.