The Hanna Barbera episodes OP is thinking of are "Island of Lost Ships" (the Sirens, though their entrancement is portrayed more as turning their victims into stone) and "Beast of Storm Island" (the Giant Cobra monster, Axor).
As for MC in the "Godzilla" franchise in general, it does crop up from time to time but usually not in ways that are especially visible. Godzilla and several other monsters are controlled by Evil Space Aliens repeatedly throughout the original series, for example (most notably "Invasion of Astro Monster", "Destroy All Monsters" and "Godzilla VS. Gigan"), but it only ever manifests as the monsters just kind of doing what they'd be doing anyway, i.e. destroying cities. "Destroy All Monsters" does also feature HUMAN characters under Mind Control, though, so there IS that, and there's likewise some Roboticization elements to be found in "Terror of Mechagodzilla", in which the character Katsura is a human-looking cyborg who occasionally goes into Robot Speak mode. Godzilla is also put under psychic influence in "Godzilla VS. Spacegodzilla", but it's ultimately a fairly minor plot point that doesn't really go anywhere.
