Joe, I found Pecos River Queen by Jack Thorpe, but I couldn't find a version with the handsome/rich refrain. I'll add the lyrics to the song I did find though.

Where the Pecos River winds and turns in its journey to the sea,
From its white walls of sand and rock striving ever to be free,
Near the highest railroad bridge that all these modern times have seen
Dwells fair young Patty Moorhead, the Pecos River Queen.

She's known by all the cowboys on the Pecos River wide;
They know full well that she can shoot, that she can rope and ride;
She goes to every round-up, every cow-work without fail,
Looking out for all her cattle branded ‘walking hog on a rail.’

She made her start in cattle, yes, made it with her rope;
Can tie down e'ry maverick 'fore it can strike a lope;
She can rope and tie and brand it as quick as any man;
She's voted by all cowboys an A-1 top cow-hand.

From http://www.texasescapes.com/MikeCoxTexasTales/Pecos-High-Bridge-Pecos-River-Queen.htm

You can hear a longer version here