Not true (at least in the US): weary (at least the adjective) is pronounced with a "we" at the beginning, while wary is pronounced with a "wear" at the beginning.

I hear "weary" spoken so very rarely that I have no idea how it's pronounced here (Indiana). I can't even make up my mind how I pronounce it, but I'd wager it's almost, but not quite, indistinguishable from wary.

They're also sometimes spelled the same. Doesn't make it correct.