if the coil melted, no doubt it did so because it was stuck on. if you just replace it without finding the culprit, it may just happen again with the new one. according to page 8 of the manual linked above, it's transistor Q15 or Q16 depending if that's your left or right flipper coil.
test the transistor with a multimeter using this - http://www.pinwiki.com/wiki/index.php?title=general#Testing_a_Transistor.2C_Silicon_Controlled_Rectifier_.28SCR.29_or_Field_Effect_Transistor_.28FET.29
if it tests bad, you need to desolder it to remove it and put a replacement one in (probably no more than a dollar or two). good luck and let us know how you get on