Good point. I have to think about this.
The atmosphere on Earth, not just the clouds specifically, does hold in heat at night, moderating the temperatures. So some heat escapes into space. I am no expert on this, but it could be that infrared light radiates from warm areas on the Earth, and the atmosphere absorbs this energy. Only some of the infrared light escapes through the atmosphere. Maybe someone who knows more about the subject than me could explain more accurately, if I am wrong.
Now on the moon, there is no atmosphere to absorb the energy from the infrared light coming off the warm parts of the moon during the night. By the same logic as on Earth, infrared light is produced by the warm parts of the Moon. But without an atmosphere, the light just goes off into space. That effect might cool the dark side of the moon. This is just what I think would happen, I'm not saying this is backed up by anything, its just a hypothesis. It would be nice if someone else could confirm this or tell me I'm wrong. I'd do some research on it, but I have an ecology test to study for tommorow morning.
Also: About it being warmer on cloudy nights: Maybe it is a little warmer on cloudy nights, but it also gets more humid, and the pressure drops a little bit. So sweat doesn't evaporate as fast, and you feel hotter. It might be a combination of a few different factors.