That's a kind of hunting I'm familiar with.
#1. Think FAST. Fast on target, fast followup shot. An O/U, with two chokes, can be a very good choice, especially since O/Us have tang safeties that you can disengage unconsciously when shouldering. My favorite gun for that kind of hunting is an old 20 Gauge 26" Ithaca SKB that I picked up for 500 bucks. It weighs less than 6.5 lbs., and it just works wonderfully for fast-moving wild-flushing birds. It is way too old to have screw-in chokes, though, so it's not a turkey gun with IC/M fixed barrels.
#2. Don't
buy a pump gun specifically for that. No matter what anyone says, just don't. Especially without a dog, expect a lot of really fast shots with very short windows to hit the bird. 2 or 3 shots gives you much more chance of hitting something than 1.
#3. If you buy a gun, get a 20 Gauge. Contrary to the belief of people who are satisfied with not hitting many birds, your average 12 Gauge is not a "do-everything" gun. For what you want, it's pretty close to a "do-nothing" gun, because it's too slow on target. It's fine for pheasant and turkey, but not the other birds mentioned. A shotgun is kind of like a golf club: size and weight have to match the speed of swing.
#4. If you want to start with your 870, get a 26" barrel, not a 28". A 28" Express is an absolute slug (I have one and I figure it'll work fine if I ever want to shoot waterfowl again). And expect to be craving a new shotgun really, really soon.