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Very possible, but the local Autozone has a cute girl working there. Small on the hoots but very cute ***. Oops, this is off topic material.
Being a betting man I would guess less, much less, than 1% would have drilled their diffs for a drain plug. Either pump it out or separate the pumpkin from the batwing.
You say you know the fill plug is crossed, did you try a little heat on it, are you using a 3/8" drive ratchet? How many miles on the car, if it's low I would seriously doubt that plug has ever been out.
Sometimes when you try to remove bolts, plugs or whatever and you'd swear Godzilla put them in. I just went thru this with brake calipers. Tighter than a MOTHERFKr, a little heat later and they came out so easy.