Unfortunately, it does get pretty complicated.
The problem with much of the West is that snow/rainfall is very seasonal. Since relatively little rain falls from late Spring to Autumn, many of these areas are indeed very dependent on snowmelt from the mountains to keep streams flowing during the drier months.
And while a warming climate means more atmospheric water vapor and so more precipitation overall, that's not true everywhere. Indeed, in most continental interiors, the increased temperatures are expected to lead to increased water loss from the soil and from surface water, and
decreased precipitation.
Most of the American West and South is expected to become substantially drier as the climate warms.