20g buckshot is just as effective as 12g buckshot. Knock down power is bull****. Shot placement is king. I won't go into how dumb a pistol grip is as we have had that argument more than a few times.
I would recommend a good 870. As has been said you can find used police guns for a great deal and most have not been shot a whole lot outside of dept yearly qualifications.
Whatever route you go with. Once you buy it, shoot it. A lot. put at least 200 rounds through it at target from 3-4 feet to 50 yards minimum. Find out how it patterns at different distances. Learn how to reload it, quickly and under stress. Shotguns are like revolvers. Short on ammo and slow to reload. So get good at it.
And some of the best advice I can give you. Get to a class. Run the gun, find out your weak points, and fix them.
And please ask any questions you may have. The only dumb question is the one not asked.
Mike