remove the cover and see what the insides look like. a bad diff with usually show clear signs like broken gears and parts, chunks of metal, etc. that will tell you what exactly is bad...like spider gears, cracked carrier, ring gear, etc.
the internals have absolutely nothing to do with your wobbles. also, there is no "slight" death wobble. DW is characterized by a loss of most steering control due to severe harmonic steering movement & vibrations, requiring you to stop the vehicle in order to regain control. you either have DW, or you don't. it sounds like you have slight harmonic steering feedback...not DW. either way you need to identify the cause - alignment, tire balance, worn track bar, worn steering components (ball joints, unit bearings, tie-rod ends, etc).