Jim,
Check the voltage at the solenoid then at the switch and last at the battery. You should see ZERO voltage drop if your electrics are in good shape and wire is right size. A drop of .1 or.2 volts is probably ok at the solenoid as compared to battery voltage.
FYI your battery voltage should be about 12.6 at rest. Lower than this could indicate a battery that needs to be charged or replaced. Does the coil seem warm in the morning, at anchor when you make coffee? Weak or discharged battery could be the root issue.....
I am suspecting you may have poor connections in the wiring to the solenoid, perhaps on the other end at the switch. The wire itself may be going south, if it has turned black or green inside the plastic jacket. Strip back a few inches and see what condition the wire is in.
Low voltage will increase current flow which makes heat. More heat makes more resistance, and so on and so on......Either way, excess heat is a short route to early failure.
Coils will sometimes be a little warm when energized but not hot. You may also have a new defective one. Check to make sure the coil is rated for 12VDC and that it is a not mis-labled AC coil. Also compare the current draw with the old coil. It may be the new one draws more current, hence the wire in your boat may be a little small, causing high current draw from high resistance...
Chris
E 34 Flight Risk