Dear WhiteHenna,
You are correct.
UAE is NOT a member of the Hague convention.
What you need to do is go to the Bulgarian Embassy in UAE and request a "consular legalization"
For a list of member countries please visit:
[url='>http://www.apostille.net/member-countries-of-the-hague-convention.html[/url'>
Hope this helps,
Luis
apostille.net
New York