Learn English – What’s the saying that those who scream loud aren’t the most dangerous ones

I have a half-answer. I have seen that phrase (or one very similar) used, but not to mean "if we involve someone in other activities, he/she will stop what he/she is usually doing".

I have seen it used to mean, "Don't talk with your mouth full." And there is really no other way of saying it that I know of.

One example source (Word doc; see page 8):

Gdy pies je, to nie szczeka, bo mu miska ucieka
Word-for-word translation:       When the dog eats, he doesn’t bark, or his food will run away. Meaning:       Don’t talk with your mouth full.

I think the phrase is an aphorism rather than an idiom. And as translated it is perfectly clear.