Hi, NJ_bird-
You say this is NOT the song by Nick Cave or Blind Willie Johnson, but the lyrics you furnish match recordings from both of them. So, I take it that your song is a different version of the song recorded by Cave and Johnson. When we cover a song, we cover the whole enchilada, so I'm going to start by posting the Cave and Johnson lyrics, to give others help in finding the song you seek.
I did not find any song titled "City of Refuge" on any of the Chad Mitchell recordings, including Singin' Our Mind - maybe the Chad Mitchell Trio (CMT) used a different title, but I have most of the CMT recordings and the song doesn't sound familiar to me. Then again, I'm proved wrong at least once a week. And yes, I've been proved wrong again - (click here).

CITY OF REFUGE
(Nick Cave version)

You better run You better run and run and run
You better run You better run
You better run to the City of Refuge
You better run You better run
You better run to the City of Refuge

You stand before your maker
In a state of shame
Bacause your robes are covered in mud
While your kneel at the feet
Of a woman of the street
The gutters will run with blood
They will run with blood!

You better run, you better run
You better run to the City of Refuge
You better run, you better run
You better run to the City of Refuge
In the days of madness
My brother, my sister
When you're dragged toward the Hell-mouth
You will beg at the end
But there ain't gonna be one, friend
For the grave will spew you out
It will spew you out!
You better run, you better run
You better run to the City of Refuge
You better run, you better run
You better run to the City of Refuge

You'll be working in the darkness
Against your fellow man
And you'll find you're called to come forth
So you'll scrub and you'll scrub
But the trouble is, bud
The blood it won't wash off
No, it won't come off!

You better run, you better run
You better run to the City of Refuge
You better run, you better run
You better run and run and run
You better run to the City of Refuge


Source: http://www.lyricsmode.com/lyrics/n/nick_cave

---

from secondhandsongs.com:

The Originals book says: originally a gospel phrase "Run! Run! Run to the city of refuge, Run! Oh Run! Or else you'll be consumed" And then "You better run" (trad.)

I wonder what the difference is, but there little information on this.

First recording possibly in 1923 by the Wiseman Quartet or Sextet. Release information unknown, but reissued on the Document label.

Other early recording on December 5, 1928 by Blind Willie Johnson as "I'm Gonna Run to the City of Refuge" released on Columbia 14391D

Also recorded on August 23, 1939 by The Norfolk Jubilee Singers as "You Better Run" and possibly released on Decca. This version was possibly the inspiration for Elvis Presley's version recorded March 31, 1972 (called "You Better Run").

In 1988, Nick Cave used similar lyrics in his song "City of Refuge", so I guess this would be a partial cover.

In 2006, Grant Lee Phillips covered "City of Refuge" by Nick Cave & the Bad Seeds.