Megadroughts are not necessarily large in extent. It is unlikely that they would affect global cooling. But global cooling could surely result in local changes in climate. Or, more directly, I imagine the energy involved in a megadrought system is possibly several orders of magnitude smaller than that of the global climate.
Yes they seem to be in principle local phenomena, so will tend to be the result, rather than the cause, of global effects. It seems that the El Niño/ La Niña oscillation may be made more extreme under climate change scenarios, in which case we may see more megadroughts in various parts of the world in future: https://research.noaa.gov/article/A...ure-of-ENSO-under-influence-of-climate-change