have the battery load tested. Just because it has 12 volts that doesnt mean it will handle the load. Load test battery, clean and tighten all connections on battery and starter. try it again. If it still clicks rapidly, remove the starter and take it to a starter repair shop and let them have a look at it.
You might want to check the voltage to the flag terminal on the starter solenoid. It should be no less than 7.5 volts. Check this both with key on and cranking to see if there is too much load. Load testing the battery is good. Need to make sure you have good voltage first. It could be a sears issue with a relay. They use them instead of fuses now.