So the question isn't "do I need a shotgun" but does it need to be a 12 gauge vs 20.
As stated - the biggest issue with 20 gauge is a reliable supply of the right ammo. Bad guy isn't going to notice what you shoot him with.
The most important thing with shotguns is technique though and shooting a 12 isn't that much worse than a 20 if you know what you are doing. My mom shot skeet with a 12 for many years and my wife will sun a load or two though mine no problem.