test voltage at the relay and see if your getting battery voltage
i will be very suprised if you do.
when the battery is charging off the alternator the voltage must be higher than battery voltage, this in turn is what the circuits in the car are designed to run at.
when you turn the engine off the voltage will be lower and therefore everything appears to run slower, this isnt a fault its just the way it is.
age makes it worse with the forementioned resistance and other age related issues.
modern cars are better regulated (power operating things is regulated to 12v irrespective of alternator output, this is why its not uncommon for modern alternators to output 16volt to charge a battery but not effect the rest of the cars electrics) they are ecu controlled with a myriad or relays sending power along the shortest route so losses are minimalised.
this can be replicated on old cars but its probably worth just living with the quirky electrics