That's what your looking for and check the coil.

If the flipper is blowing fuses, i'd say the coil is likely bad. Disconnect the power wire from the flipper coils, and check resistance across the coil and compare to the working coil - be sure to use the correct polarity, as the diode will always have near-zero resistance in one direction;
The smell was not from the fuse going (a blown fuse won't smell - it's contained in that glass tube!), it was from the failure that caused the fuse to blow.

Now why did the coil go bad while holding the ball on the flipper? I'd suspect that the EOS switch wasn't making contact, so the high power coil was left on for the whole time - if you had thought of it at the time, you would likely have found that the coil was hot enough to burn you (do a visual inspection of the coil, see if it shows any signs of overheating) - at least if the problem was in the coil - of course, it could be in the drive circuits for the coil too.

It sounds like you may have overheated the coil and it is now shorted. It could also be the transistor that drives that coil, but the fact that you smelled something when opening the coin door makes me think of the coil being the culprit. Technically this shouldn't happen, but when you hold the ball on the flipper there is current flowing through the coil, which will heat it up. If you hold it long enough, things are going to get hot.

Do you have a multimeter? You'll need one to check the resistance of the coil, check its diode, and to check the transistor that drives the coil.

Using a Multimeter check the resistance of the coil (game turned off). It should be above 3.8 ohms. If it's not disconnect one end of the diode and check again. If it's still low, then the coil has probably shorted. Also, you may have blown a transistor, and possibly it's pre-driving transistor.