Or not....
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There is only power to one terminal in the selector switch plug (about 8 volts)

If the battery voltage is now 12.5v because you replaced the battery.....right? and are getting 8v at the plug means you have a bad connection probably due to corrosion from sitting. Continue to check the voltage upstreanm till it matches closly the battery voltage then you will have found the point of the bad connection.