A stock V6 car would have a standard differential, so the spare tire wouldn't have cause a problem with that. I would be checking the u joints first. That is a much more likely problem than the gears. If you don't find a bad u joint then check the pinion nut to see if it is tight. You should be able to put about 125 to 150 foot pounds of torque on it without moving it. Then check to see if the pinion has any up and down or in and out play. If you get any movement at all other than rotation then you have a pinion bearing problem.