Meaning of "I shut my eyes to hold my breath"

Solution 1:

It doesn't mean anything. The lyrics are deliberate nonsense, full of contradictions.

Consider :

It rained all night the day I left, the weather it was dry;
The sun so hot I froze to death—Susanna, don’t you cry.

The weather is both rainy AND dry. A contradiction. The sun was hot, but the singer froze. Another contradiction.

I jumped aboard the telegraph and traveled down the river,

How is it possible to travel on a telegraph? It isn't possible, it's nonsense.

I shut my eyes to hold my breath

There is no connection between shutting eyes and holding breath. It's a non-sequitur. Another piece of nonsense. It doesn't mean anything, and that's the joke.