It is unlikely to be the coil. Where the plunger rests against the bracket (the back of the plunger with a round flat piece on it) there should be a rubber insulator grommet. If that is missing and the metal plunger is resting on the metal kickout assembly, it could cause the problem you describe. Get some felt dots from a hardware store (they sell sheets of felt dots that are various sizes under an inch) and put one on there to replace the missing rubber "rest" and see what happens.
If the rest is not missing, move the plunger by hand. If it is binding, find out why and replace whatever is out of shape, or clean out whatever is in the way.
If it moves smoothly, it is probably an electrical problem. Do you get full voltage on the coil when it is sitting there not trying to kick?
Ken