I think that she is not egg bound if she is pooping normally. If she has some salpingitis or egg yolk peritonitis with masses of internally laid egg material, those can also put pressure on nerves. It is hard to know for sure.
Since she has a prolapse and is a bit dirty, I would also place her in a pan of warm soapy water to soak and clean her up. Always keep the prolapse moistened with something like vaseline or hydrocortisone cream.