I absolutely understand and agree that you need to find a solution rather than just give in and let her be off line all the time, but I somehow don't think she's actually sulking, but rather that she does have a problem with the lead.
Humphrey went through a phase of racing to his bed and refusing to move everytime we started to get ready to go out. We finally concluded it was the harness he had a problem with as as soon as we went back to the collar again, he was fine.
I wonder if you tried to make being on the line as pleasurable for her as off by playing some sort of game - you could hold a ball on a rope for example and play with make a game of jumping over a tree log, hide a few treats and let her find them and make a big thing of it -just try and generally make the walk a bit more exciting than just walking.
Is there anything that could be spooking her and making her want to go back home? Noises, smells, etc?
I think if it was me, that I'd try and make her walk on the line a bit more often for a little while, - in all areas, beach included, just so that she gets used to not being able to run all of the time. Pulling is something they all seem to do unfortunately, - irrespect of whether they mind the line or not, the only solution to that is to try and curb it a bit by training (although I wish you luck with that, not many on here have non-pulling cockers!!
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Don't know if any of this helps, I'm no expert, they were just thoughts that crossed my mind but hopefully someone else will have some suggestions! Best of luck - keep us posted!