Yes, it's another pedantic pet peeve!
I hear more and more people saying they're "weary" of something when what they mean is that they're cautious, suspicious, or distrustful of it.
You're looking for "wary", or maybe "leery". "Weary" means tired. If you say you're "weary of" something, it means you're bored or dissatisfied with it.
Go on. Look it up. Any dictionary will do.
A plague of insomnia upon people who misuse the word "weary"!
p.s. it's my second wedding anniversary today! I can hardly believe it. I am definitely not weary (or wary, or leery) of Mr E Man.