The cause of death when a Cormorant swallows its prey alive is, in all probability, suffocation.
If not, the acid in the stomach compartments in the cormorant would interfere with oxygen transport in the gills.
Pythons, boas and other constrictors kill their prey before they feed on it, by constricting and suffocating it before they swallow it.
In other types of snake, the prey suffocates once it is in the animal’s stomach.
It is a combination of suffocation and drowning.