You are right that water is a green house gas. You are wrong to assume the only water in the air is clouds. Clouds is basically water condensation. That happens a bit less in warm air; but the water is still there. You need cool air for condensation to happen. Warm air can hold more water in gas form.
Clouds reflect light and infrared radiation from the sun. Less clouds means more of that heat gets absorbed and then trapped by green house gases. Like water.