I just want to tell you (although I feel badly for doing so) that when we had two birds do that last year, both ended up dying before I could figure out what the problem was. We examined one for possible egg binding, but couldn't feel anything. They were both RIRs and laid very large eggs, so I don't know if that was the problem or not. They were walking very slowly with tails down and had very poopy bottoms.
Now we've got another one who has developed a very dirty bottom--but she's a poor layer (Sussex) and they're all off laying right now anyway, due to the lack of light and being older. So with this one, I don't think it could be egg binding, and she's not really acting sick anyway.
I'll try to follow this thread but may start my own.
I hope things work out okay!