I'm going to assume that quail hunting in your area is a lot like it is in mine. AFAIK that is the case.
Quail is a short-window bird. They live in rough, brushy, rocky hill country. They pop up fast, and they fly away fast, and they drop behind cover fast. You have 2 seconds to shoot quail here. When you hear 'em, you shoulder, when you see 'em, you fire. Fast point-shooting is the name of the game 75% of the time. You often don't get a second shot, and you seldom get a third.
Quail here is not some pleasant stroll through flat country. It's challenging, fun, and delicious if you get some!
So, criterion #1: the gun must shoulder quickly, but not be so "whippy" that you overswing while moving fast. That argues for a 20.
Criterion #2: followup shots. AFAIK Arizona has a 3-round limit. This means that a semiauto offers you little or nothing over an O/U, in the conditions described. I think you're better off with IC/M so you have an open pattern followed by a tighter one as the bird flies away. You won't get to use a Full choke anyway.
28 Gauge may be nice, but 20 Gauge ammo is much cheaper and far more versatile. If you are going to load your own (I do), then you can load 3/4 oz. in 20 if you really want to. 20 Gauge guns tend to be available more readily and at a lower price on the used market. And they can still be used, loaded with #5's, for pheasant. Furthermore, some 28's are too quick and light for our quail. You're moving so fast you can overswing when you shoulder. And others are built on 20 Gauge frames anyway, so why pay extra for slightly smaller bores, when there's no other difference?
I'd say a 20 Gauge O/U for quail in our rocky high deserts, if I'm going to recommend something. Or shoot what you have, and decide for yourself before spending cash.
Doves, on the other hand. are a lot like shooting trap or skeet, but with someone else calling "pull." They sit on wires or bushes, then they up and fly by. A full-sized 12 Gauge semiauto can be nice for those, for the swing provided by a longer, heavier gun. But doves also can "jink", or change direction suddenly, rather than flying in a predictable line. A lighter gun like a 20 O/U can be good for that.
Bottom line for doves is, shoot whatever you want. Switch off for fun and challenge. They don't stay so close to the ground (usually), or fly fast and drop, like a quail. So IMO the gun is less critical. I've missed having even one shot at quail because I had a heavier 12 pump. They were gone before I had the gun up! With doves, I can't remember that happening. Some guys shoot .410's, for the challenge. It doesn't matter so much.
I'd be reluctant to buy a special gun just for doves. I did get a gun just for quail, though, a 20 Gauge 26" O/U, fixed IC/M barrels. Dove season is a good time to get proficient with your quail gun -- quail is the good meat anyway!
You CAN use a .410. But I think a 20 is a good all-around gun, and the first it makes sense to add to your collection.
