Your 357 mag is just fine if your going to be up close working with dogs or stand hunting them but use hardcast loads or maybe a federal 180gr SP that a think corbon has a 180gr HC that i have used for many many years on hogs. Just place a round in the head or shoot for the shoulders. It will break bone and still may pass thru. I have around 180 hogs down with that bullet. Both factory and handloads. It runs around 1200 fps is controlable and just plane works .
If you are going to have a bit of distance for a shot take your 20ga and use remington buck hammers slugs unless its rifled then what ever it shoots well. Leave the 410 at home and the 12ga is not needed.
This looks like a good oppertunity to buy a new gun. I would go with an AR in a 6.8 or 458 Socom. The 357 is a good sidearm but not what I would want as my main weapon. A 12 gauge with slugs will do, but you will be limited.