I have personally not found a bird that CONSISTENTLY lays a thin-shelled egg to be due to molting...molting birds typically do not lay at all. However, if she thinks it could be due to molting, the only way to know for sure is trap nesting, as you indicate. I would trap nest each bird for a few days to determine who it is, then I would separate her and watch. If she was on my farm, she would be culled if she continued laying a soft shell egg consistenlty. JMO.
I quite agree. Molting birds usually do not lay, as you say. But in my experience, sometimes, in the process of coming back into lay, the egg-laying apparatus misfires, and sends out an egg either with a thin shell, or none at all. It usually resolves itself in a day or two.
If I found one who was laying such eggs for more than about a week, she would indeed be culled. No room here to feed unthrifty birds. Part of why I raise this breed (aside from the emotional/historical connection for me) is because I expect them to earn their keep by providing eggs and meat. Any bird not capable of doing so is culled.
Culling for vigor and health is crucial, IMO. I seldom if ever use antibiotics or other medications, only having given antibiotics to a very expensive bird I purchased from someone else. As it was, she wound up unthrifty, and I culled her anyway.
Dr. Hatchet is a poultryman's friend when it comes to sick birds. But I recognize not everyone, especially those who are newer to poultry, with birds they consider pets, is as ruthless as some of the rest of us when it comes to dispatching the weak or ill.
And no one should be turned away, IMO, when seeking advice for such birds. That's what this forum is for, right? To help each other?