I just returned your PM, but I thought I would post the same thing here just in case anyone else on the forum had something to add. I also noticed your last post that you felt resistance when you pulled. I think that is because the egg membrane is drying out and kind of stuck. So I think it makes sense to try a warm bath or flushing the vent.
Here is what I wrote in the PM is anybody has something to add:
Sorry about you hen. I checked out your thread and I am glad she is pooping and drinking. I think she is passing an egg that did not develop a shell. Here is one thread that could be helpful:
https://www.backyardchickens.com/forum/viewtopic.php?pid=3609985
If she is acting normal I wouldn't worry about giving a systemic antibiotic, but you might want to flush her vent with warm water mixed with some iodine (mix so it is the color of tea) You can use a syringe or even one of those little plastic craft bottles with a pointy tip.
If she recently started laying, don't worry too much about that part in the linked thread about egg yolk peritonitis. Younger pullets sometimes lay unusual eggs. But it never hurts to add a bowl of oyster shell to ensure everyone is getting enough calcium for strong shells.
Of course this has to happen before you are going on vacation - isn't it always like that?
If you can get that stuff out of her and she is acting normally, I find the best time to return them to the coop is after all the hens have started roosting. That way they will wake up the next morning and act as if she never left the coop.
I hope this helps and she is back to normal soon.