Pigs with shorter snouts will do less damage, like kunekunes, but we have 3 and they do still churn things up.
I wouldn't put a ring in their snout though, I think it's cruel.
If you had room to rotate your field, say with an electric fence, you could just put up with some damage, move them over and let the land recover.
Pigs are lovely and worth the mess!
J xxx