You should check the switches that trigger that coil -- the Up Post and/or Down Post switches (whichever coil is applicable -- this is a two-coil setup, IIRC, right?) ... make sure they all have a reasonable gap in the switch. If the coil is trying to be 'on' full-time, that could mean one of the switches is stuck closed.
If that doesn't resolve it, then I'd suggest disconnecting one of the wires off the coil that's getting hot, and see if they game operates normally otherwise, without that one coil connected. (Obviously that coil itself won't work, but we can at least check on the health of the rest of the machine.) The game should still play normally without it (albeit without the post feature), so if all else is good, then at least we've isolated the issue to be directly related to that coil (and possibly its transistor and/or associated wiring/switches/connectors).