We use "swept away" to describe being carried off suddenly, by a strong force. It is a rather dramatic expression. Typical examples are related to water sweeping someone away; for example: "The boy held tightly to the ship, terrified that he would be swept away by the violent waves." English-speakers would realize a boy was on a ship, probably in a storm, and might be carried off the boat by a huge wave. He would be lost at sea. The example you provided involves emotions (fear) sweeping someone away, causing him to lose self-control. You can think of it as being carried away from reason (logical thinking) by fear. The result would be making panicked decisions instead of reasonable, or wise, decisions. Hope that is clear enough to help!

It is an idiomatic expression, but it usually means to be overwhelmed with emotion, good or bad. Literally, to be swept away means to be destroyed completely, like "The hurricane swept away my house". Well, in this expression, it is a similar meaning. You are almost completely taken out by some strong emotion, like love, or really pleasing music, etc. Ex. 'When I saw her for the first time, I was completely swept away.' - The girl is so beautiful, that I was overwhelmed with feeling.

The feeling of wanting to run was almost overpowering. In other words, Bambi wanted to run but managed to resist that urge.