When environmental lapse rate is

more than dry adiabatic lapse rate,

the atmosphere is said to be in

The rate at which adiabatic cooling occurs with increasing altitude for wet air (air containing clouds or other visible forms of moisture) is called the wet adiabatic lapse rate, the moist adiabatic lapse rate, or the saturated adiabatic lapse rate.

There are two types of adiabatic lapse rates...wet and dry. (wet is also referred to as saturated or moist) To the extent that the cloudiness your question refers to represents saturated air, then no, the wet adiabatic lapse rate would be lower (approximately 1.5C/1000') than the dry adiabatic lapse rate (approximately 3C/1000').

environmental lapse rate involves the actual temperature of the atmosphere at various heights.

adiabatic cooling is the cooling of air caused when air is not allowed to expand or compress.

less than the wet adiabatic rate.