The main problem in The Witch of Blackbird Pond is that Kit doesn't fit into the society and her differences make others accuse her of witchcraft. The solution to this is for her to recognize the requirements for her to fit into her new society and adapt to life in Connecticut. However, her solution for her unhappiness in society is to find a way out of it.
Kit lives a completely different life in Barbados than she does in Connecticut. This is clear when she arrives in the state and leaps into a river to recover a child's toy; the water is shockingly cold, and people aren't pleased that she knows how to swim. It's one thing that signifies someone might be a witch. This foreshadows that later in the story, Kit will be put on trial for witchcraft.
She tries to adapt to her new life but isn't able to do so at first. It takes time to recognize what society expects of her. It isn't until her friend is arrested for witchcraft and then she, herself, is put on trial that the dire nature of her circumstances is clear. The solution to the accusation of witchcraft, however, was integrating herself into the community; she helped a girl learn to read, and the girl testifies and saves her from being punished as a witch.
Ultimately, though, Kit gains real freedom when Nat Eaton returns. She's in love with him, and he has his own ship. It's clear that they're going to be married and leave together, sailing wherever they please around the world. This will allow Kit to live the kind of life she wants and keep her free of the restrictive Puritan society.