Cooling would be from volcanic ash, aerosals, sulphur gas... in the short term, also depending on the type of volcano, as a lava flowing one would produce less of them than a pyroclastic one like Mt St Helens.
Should the lava keep flowing over time, like those that produced the Siberian or the Deccan Traps (the former spanning millions of years and causing the Permian Extinction, subsequently giving rise to the dinosaurs, the latter hundreds of thousands, and witnessing their demise), carbon released into the atmosphere causes a temparature rise.
The problem with a global winter or cooling event 65 million years ago is that far more sensitive and prone creatures like amphibians as well as crocodilians, etc, survived. It's not like tropical plants such as tree ferns and cycads just waited out a million years then started growing again. Crocodilian sex is determined by temperature, cooler producing more females, warmer, males. So dinosaurs going extinct due to rising temps might have been from similar.
The moon exerts tidal forces on the mantle, the churning provides for volcanism and thus the release of CO2, etc. Remove the moon, and plate techtonics and volcanoes, as well as Earth's magnetic field, cease. CO2 (and other gases) is being expelled into space at a significant rate (this was discovered a few years by infrared images taken by satellites of the edge of the atmosphere from the darkening side as the Earth turned). So no volcanos means cooling temps over time,,,