It's poetic language and it's not easy to be certain. "To draw" can mean "to pull." A horse draws a wagon. We often use the word "draw" with curtains and shades. "The sun was shining directly through the window, so I drew the shades to keep the room cool." I think Steinbeck was "mixing his metaphors" here. When he says the day had "drawn" a pale wash of light, he means that it appeared to be moving or progressing, covering more and more of the sky, like a curtain or shade. By calling it a "wash" of light, he means that, like a thin layer of water paint, the appearance was that the sky was being changed and lightened, but that nothing was being hidden.

Another common (but less obvious) use of "draw" is "to pull". This is the meaning in your example. Your understanding of "wash" is correct.