What sometimes happens is that when the broody gets off the nest for her daily constitutional another hen gets on the nest to lay an egg. The broody then goes to the wrong nest. Some broody hens are more prone to that than others. When that happens I just put the broody hen back on the correct nest and usually get good hatches, even if the eggs get pretty cold to the touch.
But it is frustrating. The problem with moving a broody to another nest is that she might break from being broody, which you don't want. If you can build a pen around her current nest so you can feed and water her but keep her from leaving the area and keep other hens from getting on that nest, your problem is solved.
A lot of people have great success in moving a broody hen without breaking her, though occasionally it doesn't work. Some of the tricks I've learned to improve you odds are to make the new nest fairly dark. A dark nest appears to be "hidden" which the broody likes. I make the new nest so I can lock her in it for the first day I move her, letting her out at the end of the day just before dark. That's not cruel, broody hens are used to staying on their nests like that. I typically move them at night, using as little light and commotion as possible. Make sure the hen cannot leave the new nest area and go back to her old nest.
Good luck!