An entire diff center will be the way to go.
But I'd still tighten the pinion nut, drain the diff oil and look out for metallic particles as well as feel inside the drain hole for any large items. Then jack both rear wheels and spin the pinion by hand feeling for any roughness. You will need to spin it anyway if tightening the pinion nut because there is such thing as too tight.
But you might get lucky...