The carrot soup shouldn't make her poop more runny ideally. The soup contains oligosaccharides that somehow help against bacteria in the intestines that could cause the diarrhea. It is a common home remedy where I live, it works for children, dogs and also chickens.
I don't have any experience...
Oh poor girl :( I hope she gets better soon!
I'm not an expert on chicken breeds but even hybrid hens can get as old as 10 years. 6-7 years is more of a usual age for chickens though in my opinion.
To me it doesn't really seem like she might be egg bound, symptoms of egg bound hens are usually...