Breathing through my mouth can't help. The cause of stinging/watering eyes is the vapour coming into direct contact with the eyes, not being breathed in through the nose. (Unless the latter separately has an effect on most people via their sense of smell, but that doesn't affect me.)

This is easily proved by the experience I describe above: if wearing goggles stopped my eyes watering, it must be because the cause of watering eyes is something that can be prevented by goggles, and therefore breathing through nose or mouth is irrelevant.