Look at a used Mossberg 500 or Remington 870, you should have no problem finding something in your price range. Even a pre-owned pump shotgun, unless it has been absolutely abused, will be fine.
Maverick or Charles Daly should be fine for home defense use and can be had new for what you want to pay.
I have no problem with 20 gauge for home defense, and you'll probably be looking at # 2 or # 3 buckshot for that use.
If you go 12 gauge, standard HD loads are 00 buck and maybe # 4 buck depending on where you live (apartment, etc.)
The rule of thumb for 20 gauge vs. 12 gauge is that the 20 will give you about 75% of the effective power for only half the recoil. At typical home defense distances, a 20 gauge is more than adequate.
I don't think I've ever heard before that a 20 gauge delivers more felt recoil than a 12 gauge.
With any shotgun, I'd recommend a good Defensive Shotgun training course..............in fact, I know a guy