I think you'd have a pretty good chance of drawing a license for the areas around Douglas, especially being a nonresident. There are a lot of antelope, and like you mentioned, there would be many additional doe/fawn licenses available also. I am also assuming you know (because of your buddy's friend's experience) that most of the land around Douglas is privately owned and finding public land on which to hunt may be a challenge. Generally, private landowners in that area of the state will allow you to hunt after paying a trespass fee, which may be substantial. Or you may be lucky and find one that will let you hunt for free.
Anyway, I wish you luck. Done right, antelope hunting can be a blast. A good first step is to choose a Handi.