weary –adjective 1. physically or mentally exhausted by hard work, exertion, strain, etc.; fatigued; tired: weary eyes; a weary brain. 2. characterized by or causing fatigue: a weary journey. Leery –adjective 1. wary; suspicious (usually fol. by of): I'm leery of his financial advice. 2. Archaic. knowing; alert. Get it straight please people.
You're half right. There is rampant misuse of weary around here, but its not leery that I think they think they're using.
If people learned how to write clearly and correctly, professional editors would be out of a job. And I have no other skills to speak of.