Welcome to Arizona. Just a little note to let you know what you're getting yourself into...March 21st, 3:27PM temp in my area was 104! Yesterday, 3-22, again the hotspot in the nation, 102. It's March for goodnessakes!!! What happened to spring?
Seriously, your opportunities to hunt elk will be tough if you're after a bull, much better if you're after an antlerless elk. You can draw deer hunts with a fair degree of regularity so long as you're not after any of the buck rifle hunts conducted north of the Colorado River. If that's the case...well, you're likely to wait for years and years and years.
Good Luck on your move. One good thing about coming out in August though, you'll be arriving during what should be a nice pennant race in the NL West with our D'Backs hopefully in the mix.