12 ga won't give you anything more than 20 ga offers.
The "power" difference between the two was a byproduct of case capacity limitations in the black powder era. But we no longer have those limitations. Today, you can load anything in a 20 ga shell that you can in a 12 ga, unless you are talking 3.5" mags. (But I assume not, since if you're afraid of recoil being an issue, 3.5" shells would likely be an issue anyway.)
So don't go for 12 ga thinking it'll be more powerful.
It will simply be different.
Anything you have expressed a need or desire to do, 20 ga can do just fine.
Find the shotgun that you want, and then see what gauge it is chambered for.