The specific heat capacity of a gas is the quantity of energy that must be added to a given amount of that gas in order to raise its temperature one degree. We will discuss the specific heat capacities of dry air and water vapor in a constant-pressure process. Though it takes more energy to raise the temperature of one kilogram of water vapor by one degree than it does for dry air, it is important to remember that in a mixture of dry air and water vapor, the latter gas accounts for no more than four percent of the mass of the mixture, usually much less. Even in the muggiest air mass imaginable, water vapor can raise the specific heat capacity of the gas mixture by only about three percent. Since the amplitude of the daily temperature cycle is reduced by far more than three percent in the example you give, we cannot blame the specific heat capacity of water vapor for this behavior.
Radiative effects are to blame, directly or indirectly.
Clear air absorbs little solar radiation, most of which arrives at wavelengths between 0.4 and 0.7 micrometers. Therefore, when the sky is clear, a very large fraction of solar energy incident at the top of the atmosphere reaches the ground. If the soil is parched, whatever solar energy is not reflected skyward is absorbed, thereby raising the temperature of the ground. The air in contact with the ground heats up as well. On the other hand, if the soil is moist from recent rains, much of the incoming solar energy is used to evaporate water near the surface.