Would it be the recoil or the anticipation of the recoil? Seems like when you shoot the bullet would come out so fast out of the barrel that it would not have time to be affected but I have no idea. I am pretty new to handgunning.
The trigger on this 19 seems rather hard to pull, I have shot a few semi-autos which had much easier pull of trigger to them. Wonder if the extra tension is jerking my hand and or wrist when I pull the trigger? I have shot a S & W 44 magnum and also to me the least accurate of the guns I have shot, must be my fear of the recoil?
My $0.02:
Sounds to me like you're either 1) flinching and/or pulling your trigger finger in towards your palm as you pull the trigger. Either way, sounds like you're disturbing the sight picture just before the gun fires. You may also 2) not be seeing the sight picture correctly. Here are a few suggestions:
1. Rather than 15 yds, shoot at 7yds. Once you start getting decent groupings at your point of aim, start placing the target further out.
2. When loading live cartridges, load one chamber with an empty casing and give the cylinder a little spin before closing so you don't know where the empty is. If you're flinching, you'll definitely see it when the gun "clicks" on the empty case and your hand & gun jerk. If you are flinching, forget the .357magnums and shoot mild .38spls for now (in fact, I'd recommend that anyway for general target practice). If you can find some, try some target wadcutters, which are generally pretty mild loads.
3. Using empty case or snaps caps, dry fire practice at home. Lots and lots of dry fire practice! Focus on maintaining a good sight picture while smoothly pulling the trigger without disturbing the sight picture. Even experienced and accurate shooters do lots of dry fire practice! It'll help your trigger finger strength as well, which alone may help your shooting if you're struggling with the pull weight of the trigger. Yes, assuming you're referring to the double action trigger, the pull weight will be considerable harder than a semi-auto, which is one reason shooting a DA revolver well takes lots of practice!
4. Experiment with grip and trigger finger placement on the trigger. Your grip should be as high on the backstrap as you can get it. As far as trigger finger placement, many revolver shooters place the 1st joint over the trigger and feel this gives them more control with less strain while pulling. Pulling the trigger with the pad of your trigger finger over the trigger (like a semi-auto) seems to take a lot more effort, and you may be straining and jerking the gun quite a bit when pulling the trigger. Again, experiment and dry fire, dry fire, dry fire!
5. If you can swing it, invest in a S&W K-frame .22LR revolver, such as a model 617, 17 or K-22. One of these, in the same platform (i.e. k-frame) as your Model 19 would be an excellent "understudy". Much easier on the wallet and will pay for themselves in the long run, and lets you shoot much more, and very little recoil so you're not reinforcing a flinch. New 617s are pretty pricey, but used 17 and K-22s shooters (i.e. non-original grips/no box/some bluing wear) in decent shape can be found for less. Like your Model 19, they'll hold their value as well and are darned good guns to boot.