With the run away/escape, that would be consistent [in line] with my original thinking, that the "brother" was running away from something and into the desert, [where not many would be likely to go looking for/find him] and where he's building his own house.
As you can already see, interpreting what Leonard means when he uses any particular word or phrase can be very difficult for those who alreaady speak the language that the songs are written in! When you add in the difficulties that come with interpretation, you've complicated it still more.
I'd have to say that your guess is as good as any, with how you hear the song. I really don't expect to see Leonard clueing us in as to how he meant it.
What a worthwhile motivation for you to have started learning English.
~Lizzytysh