Important distinctions

A. Wary vs. weary.

1. Wary: the condition of  being cautious and/or watchful. Pronounced "ware-eeee"; rhymes with scary.
2. Weary: the condition of being tired. Pronounced "weeeer-eeeee"; rhymes with bleary.
• So no, you are not weary of, for example, mailing off your original birth certificate, as it's unlikely you have to do that often. You are wary of doing so, and with good reason.

B. Loath vs. loathe.

1. Loath: unwilling or reluctant. An adjective. Pronounced like "Los" if you had a lisp - with a "th" as in thick; rhymes with growth.
2. Loathe: to hate or abhor. A verb. Pronounced with a "th" as in thy; rhymes with clothe.

• So no, you are not loathe to admit something. You're loath to admit it, and no one asked anyway. Loathe is a verb, and therefore people should stop fucking trying to use it as an adjective.

C. Anyhow vs. anywho.

1. Anyhow: haphazard; as an interjection, "in any event."
2. Anywho: not a word and therefore never to be used. 
• There is no situation that calls for the use of "anywho". Ever. Don't use it. You're not Stick Stickly, and it's not 1997.

My coworkers are professionals who work with legal shit all day, and can't be bothered to nail down these fine points. Glad to clear it up for them.

What grammar mishaps really chap your hide?