Are the plungers missing? Reason I ask is because I've seen melted coil before but never anything so bad that I couldn't reuse the plunger after some clean up.

As to why they melted. - It's almost always a stuck switch causing the coil to stay energized way longer than it should, Could simply be a stuck playfield switch or could be a switch on a relay not opening properly. Best thing to do would to look at the schematic and see what switches trigger these particular coils and make sure they are all opening and closing as they should, and are not sticking in one position.

By the way - The 2 worst melted coils I've ever seen were in games where a previous owner bypassed a fuse. One had an alligator clip jumper wire which also melted and the other had a #12 solid Romex looped over it. I have no idea how that one didn't burn up completely. So I would check to make sure no one did any funny stuff to the fuse block, just in case.

Yep, I had a feeling about a fuse when you said how badly the coils were melted.

As long as you don't shorten that plunger any, you should be able to make it usable. File the edges of the mushroomed part and put a slight bevel on that end, and give it a try.

Your won't burn a coil up instantly. You can start a game, close switches one at time, and observe which ones energize the coils in question. If one switch causes the coil to stay energized, unstick that switch manually or shut the machine off and try again.