Thanks for the great advice and insight -- especially to take it apart and closely look at it. I did (for the 20th time), but this time I did as you folks said and examined the workings. I think the problem I have is the recoil spring assembly is the wrong one for this gun, or, at least part of it is. I think that it is both too long, and the plate that rests near the barrel-connector part is not concave enough.
The first part (being too long) prevents me from not being able to pull the slide back enough so the slide stop catch is pushed down by the slanted front of the cut-out on the slide. The maximum point I can slide it back is about 1/16" beyond where it catches (and, there is no buffer blocking this travel). The overall length of the assembly is 2 5/16".
The second part is what I think is causing the slide to lock back after each round. the lack of either the plate being concave enough, or the plate being held back too far by the retaining ring, is causing the end of the assembly to push on the barrel connector. I can tell that is causing a problem, both in the way it pushes the almost all the way back (making re-assembly a little tougher), and when I take out the slide stop, I can feel movement (showing it is under pressure). I think somehow this pressure is causing it to lock back after the round is fired (it does not lock back when I just feed a round into it).
To demonstrate how my recoil spring assembly is different, compare this picture of mine spring assembly (Recoil Spring P10.45) to that of one posted by another party on a different forum (recoil spring - Warthog). With mine, you can see the end of the rod extending about 3/32" beyond the plate.
Now, I just have to find a place to get the correct replacement (Brownell's shows it as a discontinued item). Thanks for everyone's input.