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Are you sure that you replaced the correct switch??? Try this: Remove the connector from the switch, install a jumper wire between the two pins on the wiring harness connector that plugs on to the switch. When you turn the ignition switch on the reverse lights should be on. Remove the jumper and see if they go off. The reverse switch is on the passengers side of the tranny. When you remove it there should have been a ball bearing type actuator on the tip of the switch.
Put the ohm meter leads on the terminals of the old switch and press the ball bearing in. If the switch is good, the meter should show cllose to ZERO onms.
O.k. folks.. The switch is just one component involved here.. There is also a 10a fuse is the passenger-side floor board panel that typically blows when the switch fails..
My money is you'll find the blown fuse, replace it, and whala, backup lights.. Don't ask how I know!