I prefer to use a single thickness of abrasive paper, and turn it over after I've done one contact surface. Doubling it increases the angle between the two contact surfaces after the job is done. You might try measuring the resistance between the points when they are closed: it should be a very small fraction of an Ohm.
Replacing the points, unless they are rather beaten-up, is a considerable expense that should be unnecessary IMO. Once they meet squarely and have clean surfaces, the next concern is the condenser. In my experience condensers fail far more often than coils, and of course are much cheaper to replace.
Just one point: are you really sure the points open just as the magnet is close to departing from the coil?
