In a cowboy song the narrator, who is riding an old paint (spotted horse), says the dogies (motherless calves) "feed in the coolies & water in the draw." Where do they feed, and where do they water?
The more common spelling of the first word is coulee (though cooley and coolie are also both found). It is a North American dialect term for a deep ravine (originally ‘lava flow’—a somewhat unusual semantic development). The OED has a slightly more detailed definition:
“In the Western regions of Canada and the United States: A deep ravine or gulch scooped out by heavy rain or melting snow, but dry in summer.”
According to them, “Sense 2 [the ravine, rather than the lava flow] appears to have arisen among the French trappers in the Oregon region”, though their citations for the ravine sense predate their citations for the lava sense by about 30 years.
In this sense, a draw (look under the noun section, sense 6) is –
“A gully shallower than a ravine” (Merriam-Webster)
“A natural ditch or drain that draws the water off a piece of land. Also, a shallow valley containing a stream.” (OED)
As far as I know, neither word is in particularly common widespread usage (I’ve never personally heard draw used in this sense, though the comments indicate that it is more common than I thought), though I expect they may be in certain particular areas or vocations.
From my own experience, growing up in the US mountain west: "draw" is pretty common, at least as much so as "ravine." I'd define it as a drainage smaller than a canyon.
"Coulee" is very common in the Northwest US, particularly Washington state (cf. the Grand Coulee, as someone mentioned), and in fact, having lived in Washington for a number of years, I think of it as a northwest localism. It tends to be used instead of "canyon," particularly of a flat-bottomed canyon with steep sides. As used there at any rate, a coulee is not a small or shallow feature!
Coulees are larger, deeper, and have steeper sides than draws. The Grand Coulee Dam was built at the narrowest point between ravine/draw/coulee walls.