That fire is a long ways from Tucson and isn't very easy to access. From Tucson you're looking at about a six hour drive to get where it's burning. There is still a ton of area to hunt in AZ, you shouldn't have any issues with the fire. As far as the draw goes, you don't have to draw for archery deer (you do for a few certain areas). You can buy a tag otc that's good for pretty much the whole state and you've got about three months to fill it. Elk, antelope, javelina, spring turkey, and some bear hunts are all draw. If you haven't already, check out the AZ Game & Fish website, they've got the rules and regs online.
Feel free to PM me if you have any questions. I'm no expert but I have a pretty good understanding of how things work out here.