Origin of Ginger

I don’t think you drink a drench. Drench is a verb meaning “to soak thoroughly”.

Although there’s apparently an alternate definition for a dose of medicine forcibly administered to an animal.

Hmmm, apparently on word origin from Old English it meant “to force to drink”, still a verb, with related form drenc for a drink. So maybe you’re almost right.

No, you don’t “drink a drench” in modern English. But it seems pretty clear that both words had the same root somewhere, meaning water or wet. “Drink” is still used as a euphemism for a body of water.

Yes, the words are related—indeed, they were dialect forms of a single word until relatively recently.