Sounds like you're forcing the hammer into a faux half-cock, which can damage delicate mating surfaces.
When lowering the hammer on an unloaded gun, the trigger must be fully to the rear, or fully forward (on a loaded gun, it needs to be fully forward, with the finger off the trigger). The tolerances in the action are pretty tight, and in some (but not all) guns, releasing the trigger while lowering the hammer can bind the action in a half-cock the gun wasn't designed for.
Check out the series of pics below. I made them specifically to explain the faux half-cock. Study pics 1-4 for normal function, then pics 4 & 5 to see how you can jam the action into a half-cock. Pic 4 shows how the action can half-cock with the trigger fully forward if the sear's too long, but you can also jam the action this way with a proper-length sear by letting the trigger out while lowering the hammer if the tolerances of your particular gun are "just so".
Being able to get your gun into half-cock doesn't mean there's anything wrong with your gun (yet). Some guns can do it, some can't. But as you can see, the gun wasn't designed for it, and you can damage it by doing it.
EDIT added: If you do jam the gun in a half-cock, don't release the half-cock by simply pulling the hammer or trigger individually. Yes, it will release the jam, but this is where it's most likely you'll damage your gun. Instead, un-jam the same way you jammed it, gently pull back both the hammer and trigger simultaneously. The movement should feel smooth. Once done, allow the trigger to fully release forward, then lower the hammer.
Thanks - I will take the side plate off, and try to see what is going on so I learn to understand it. I've never heard of faux half-cock before. The more I learn, the more I find out I have yet to learn.
If you watch the video of what I'm doing now, I don't think it's what you refer to as faux half-cock. That doesn't mean Will or I didn't do that unintentionally, while we were trying to figure out what is wrong. It should be clear in the video that I pull the trigger all the way, the gun "fires", and I simply release the trigger slowly - there is a specific spot where something "binds".
There's also the possibility that this is what's been wrong all along, that ages ago someone didn't do this, "which can damage delicate mating surfaces". My totally uneducated gut feeling is that what you wrote here describes what I'm feeling, and there is some kind of mechanical interference between parts.
After cleaning the gun, we dry-fired the gun many times, both single action, and double action, and something felt wrong - the trigger didn't want to return all the way forward. Will tightened up the screw against the main spring, and it got much better, but this "bind" was still felt. Last night I tightened that screw all the way, so the main spring now looks like the photos I've seen. The bind happens after "shooting", hammer fully forward, while releasing the trigger. If you release the trigger normally, you barely notice the issue. If you release the trigger as slowly as you can, it gets "stuck" while moving forward.
Last week, back at the gunshop, my totally uneducated guess as to where the problem was coming from, was the area inside my red circle over the photo, and the same parts you've shown in your very last photo, at the bottom right. I'll try the new springs, but suspect the problem all along has been that one or more of these parts got damaged, long before I got the gun.......