Hi J.,
We had problems with the swing as well. My son ONLY took naps in the swing and then did nightime in his crib. It took us 6 months to figure out that he really needed to be napping in his room in his crib. It was so hard to make that transition.
You can totally do it slowly (1 nap in the crib/bassinette) for a week or a couple of days and then move it up. He's still a good sleeper, so I don't think being in the swing for the first 6 months of his life scarred him, but it was hard on us waiting that long to make the transition.
At least when she is laying flat, you can use a sleep positioner (home-made by rolling up a blanket or store bought) and allowing her head to get "a new shape."
Good Luck,
B.