A lot of egg binding occurs due to two eggs in the works. The first time I saw this was with an EE hen years ago. I treated her with a warm soak just as you did. She produced an egg. I wanted to give her time to recover from that and to rest, so I left her in the crate. An hour or two later, I came back to find she had passed another egg, this one without any shell.
Since then, I've had more hens suffer egg binding due to ovulating twice within a 25 hour cycle. Almost always one of the eggs lacks a shell, causing difficulty passing it. I've learned that the minute I suspect egg binding to give the hen a calcium tablet, 400-600mg, preferably with vitamin D3 plus minerals. This helps them pass the eggs more easily.
For stubborn cases that drag on, I use a heating pad on the lowest setting with a damp towel laid on top of it. The hen is placed on top of the damp warm towel so the moist heat can further aid in passing the egg. Lubricating the vent inside and out with a little vegetable oil or coconut oil helps, too.
