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Both ends of the worm gear are supported in a bearing with a ball bearing inside it.
You may be able to lift the free end of the shaft out with the one bearing enough to then slide the shaft out of the other bearing without having to remove the coupler retaining plate.
Turn the shaft by hand in both directions and watch the mesh of the worm gear with the worm wheel on the axle. Does it mesh properly? What is the condition of the worm gear on the axle?
Larry
Larry