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Hello and thank you for asking. The next thing to do is check the voltage at the motor while the winch is running and then under load. This will check for a voltage drop at the motor that would cause the exact symptom you are describing.
To do this test set you meter to DC volts and put one probe on each of the motor terminals. Then have someone else run the winch. With no load you should have almost no voltage drop. Thus you should show about 12.9 volts at the motor while it running with no load. Then load up the winch and the voltage should hold above 11.9 volts. If the voltage drops out then we have found the problem.
If the voltage does not drop out and you have changed the motor then I would suspect a binding on the winch under load while line pulling. This would be very rare and you would hear some squeaking from the winch. So lets do the voltage check first and go from there.