Shalom's suggestion that it was a corruption from ב׳ אדר seems plausible, although of course ב׳ אדר is itself awkward. Another guess might be that in listing months sometimes people wrote the 12 and after אדר simple added ואדר, as if to say "and Adar," like "Shevat, Adar and [again] Adar."
See Shekalim 2b-3a in the Vilna edition where ואדר השני is found. It's possible that the influence of the expression אדר הראשון ואדר השני gave birth to ואדר. Also see M.D. Davis's Shtaros, Hebrew Deeds etc. (pg 105) which includes a document from Norwich, dated 1264, which refers to ירח ואדר, so it's quite old.
It's a good topic, and I think I'll research it more!
As for why it should have fallen into disuse, it's anyone's guess. But there are numerous examples of common useages which simply disappeared for no apparent reason. Take the use of gershayim (") in abbreviations and the like. Today I think they are universally applied only in the second to last letter of a term (like רמב"ם). But well into the 19th century you could still find רמ"בם and the like. What happened and why? I don't know, but it did change.