So the Wyondette is still having periods of acting droopy and weird, and laying eggs w/o a shell. I think we've narrowed it down to an oviduct problem, as when she does lay eggs (1/2 per week, WAY below her norm) they are shell-less. They don't have the hard calcium shell around them, just the thin membrane that supposed to be inside of the shell. She has laid 3+ of these now, so it's not a fluke. They are normal otherwise (size, yolk, shape). The thin membrane holding the egg together is weird, and like a thin piece of tissue paper that holds the eggs together when gently handled, but easily ruptured. Below is a picture of the egg w/o a shell (it didn't go to waste, was fed to the dogs).
My much more experienced chicken friend thinks it is an oviduct problem, and the hen needs to be culled. I'm kinda bummer out, as I did just get her. And she is the Easter Egger's friend, they hang out together and she protects the Easter Egger when the others pick on on her. But she is droopy and sad looking a lot, so think she will probably need to be culled.