Hey J-Bar (Binks?),
Fellow cowboy shooter here. I've also been playing around with bullseye, sort of as winter cross training. While I'm hardly a 45 year veteran, I have read into it some, and can tell you what's helped me.
The first and most basic thing to realize, is that the gun starts to move in recoil the instant that the bullet starts down the bore, long before it clears the muzzle. If you want the bullets to hit in the same place every time, then your job as a shooter is to ensure that the gun starts out pointed in the same place, isn't disturbed when you trigger the shot, and then that it moves the same way in recoil each time. Small differences in grip, muscular tension, or bodily position can make for quite large shifts in point of impact. Thus the key to accuracy is consistency. If you try a few shots one way, then change something, and try something else, then shift around and try something else again, your bullets will be all over the paper. If you make your body a consistent launching platform, then they should all stay in one group. All that's left is to adjust the sights.
First is stance. The big idea here is called neutral point of aim. This is the position that allows you to hold on target with the least muscular effort, and therefore the least fatigue, and least potential for variations. Also, if your position is natural, you save time in the timed and rapid strings, as the gun lifts in recoil, then settles back where it was, without you having to fight it back into position. The way you find this natural position is to stand in something like a stance, and close your eyes. Then lift the gun. Let it sway right and left a bit, until it finds a place it wants to "hang." Now open your eyes. It probably won't be pointed at the target. That's fine. Instead of muscling it around, out of its nice, natural position, shift your feet around until it does point at the target, naturally. That's your stance. Stand as relaxed as possible, and anchor your non shooting hand somewhere, like in your pocket, or hook your thumb through a belt loop. That keeps it from swinging around, and acting like a lever to move the rest of the body around.
Next is grip. The 22 isn't as sensitive to variances in grip as the 45, since it moves so much less, but it's still a pistol, and they're just about all as sensitive as fiddle strings. Obviously, it's still a good idea to get as consistent a grip as you can. Instead of just picking the gun up off the table with your shooting hand, place in in the shooting hand with your off hand. Watch carefully where your fingers go, and put them there again next time. There's lots of argument about how hard to grip the pistol, but since consistency is the goal, I think the guiding principle should be how hard can you grip it all the way through the match? If you start out with a nice firm grip, but your hand gets tired over the course of the match and eases up, your POI is going to wander around. So, as firmly as possible, without either tensing or tiring.
Breath control isn't an area to which I've devoted a lot of attention as yet, so I'll just say what I've been doing. I take a few deep breaths before raising the gun, then exhale slow and smooth while raising it, stopping as the sights start to settle. I just extend the natural pause between breaths, rather than sucking in and half out, anything like that.
Sight alignment pretty much comes in two flavors: 6 o'clock and center hold. Although I use center hold for everything else, (cowboy, practical, defensive, hunting, etc) I use the classic target shooter's 6 o'clock in bullseye. Because the black iron sights show up clear against the white target paper, and the sharp edge of the bull provides a precise place to hold, it seems to work better. Black sights on a black bull just run together for me, and I get no definition, nor can I easily hold the same place in the center. As always with iron sights, focus is on the front sight.
There are at least three schools of thought about trigger control. Let's call them steady squeeze-surprise break, interrupted squeeze-surprise break, and trigger prep-intentional break. The first, the steady squeeze, is the simplest of them all. In it, you basically accept your wobble zone, and squeeze. At some point, the gun will fire, and the bullet should strike within the wobble zone.
The second, the interrupted squeeze, is a refinement of the first. Instead of steadily squeezing through, you only add pressure as the gun is wobbling toward the center. As it passes, you stop squeezing, but do not release. Hold what you've got until the gun starts to come back, then begin adding pressure again. This should reduce group sizes, as the gun should only fire as it's moving toward the center.
The third, the trigger prep-intentional break, does away with the surprise element of the first two. This means you have to be able to trigger a shot intentionally, without blinking, jerking, twitching, or flinching. If not, stick with the surprise methods. As to how it works, if you have, say, a three and a half pound trigger, as you're letting the sights settle on target, you take up about three pounds of tension on the trigger. When the sight picture looks right, you add the last few ounces and break the shot. As Elmer Keith said, "When you see a 10, shoot it." Ideally, you want breaking the shot to be a subconcious reaction to seeing a perfect sight picture.
As for how to bring the gun to the target, it's something else I've not given much thought to, but so far I swing up into it from the bottom. It works. I imagine down from the top would too, but I wouldn't want to mess with a careful stance by introducing unnecessary lateral movement.
The final, and probably most important aspect of the game is mental. If you don't plan what you want to happen, there's not much chance that it will. Most shooters tend to develop rituals, sort of like a checklist or mental program that they go through to prepare. First place the gun for the grip, then check the stance for NPOA, then mentally rehearse the next shot or string. By then the range commands will be starting, so load, breathe, raise, prep, and then let go and let the subconcious run things.