I think these particular types of English language crimes are some of the most forgivable. You know, the ones where the person is repeating a phrase they've heard spoken all their lives, but have rarely—if ever—seen it in writing. However, have you actually thought about the words you are saying? I know that the answer to that is probably 'no', but these are not obscure words like jibe. Wary and weary make look and sound similar, but they are common enough and have very different meanings. Getting them confused really shouldn't happen.
To be wary is to be cautious or to be on guard. To be weary is to be tired or fatigued.
Now that we've set that straight, you are not "weary of making a mistake" or "weary of that backstabbing coworker". You are wary of them! However, I am weary of hearing people say that incorrectly.