If you prefer an O/U, have you considered used?
I have an older Winchester 101 20 gauge that I really like, though I purchased a little above your stated budget. I also had an older Browning Citori 20 gauge that I liked. The Browning was an 80s gun in like new condition that didn't move on Gunbroker I suspect because of out-of-favor 26" barrels, no extra choke tubes and a poor description by the seller. I think I paid $1,000 and not sure it had ever been fired. The Citori 12 gauge is pretty heavy but the 20s can be nice.
I also picked up a really nice Charles Daly (Miroku) 20 gauge with a semi-pistol grip (Browning Lightning style) in the Cabelas used rack for around $600. This gun is the same as a Citori but without the Browning name. It was a 28" M/F and seemed lighter than most Citoris and handled great. The feel was really similar to my above mentioned 101 but I was able to buy it at less than 1/2 the price. That gun falls into the rare category of shotguns I wished I'd kept.
I've never hunted AZ but have spent more days than I could count in Southern NM hunting quail and the not firing the second barrel on inertia trigger guns has been a problem for me when the wind and sand blow with certain models. Sand, sage and chamisa (rabbitbrush) get in there and cause problems. I've not seen this problem as of yet with a 101 (mechanical triggers I think) or a Browning Citori, BSS or Miroku, which are inertia triggers but don't seem susceptible to this.
Some gun designs seem to be worse than others regarding this issue in those conditions.
If the OP could find a Winchester, Miiroku, Browning, or even an SKB for $1000, that would do just fine, I bought a nice Merkel model 8 for $600 about 10 years ago, but I fear those days are over.
