Oil on the Alternator, or the cables / rectifier, could cause it to stop charging the battery and providing electrical power to the car. I have the I-4, so I am not sure where your alternator is, or its proximity to the 'Timing Cover Seal'. I think the V-6 has a Timing Chain, so it could / may have oil on it, but enough to flood out an Alternator? I don't know.
You may need an alternator. They can test the voltage it is producing when it is running (and should be able to show you this.) You may want them to do any other electrical fixes while it is in (i.e. wiring harness, blower motor switch recalls)
That's all I have. Sorry not much help.
