Yes, Maria Feodorovna lived in Denmark after her escape from Russia. She stayed for a few months in England, but her presence caused some trouble and she moved to Denmark. I don't think that the main problem was the sisters not getting along.
In Denmark her parents old home, Christian IX's mansion at Amalienborg Palace was her winter home. After Christian IX's death the mansion was put at the disposal of his daughters, so MF didn't really live there at the kindness of Christian X. During the summer she lived at Hvidøre. Knowing that she had two places to call home gave her much comfort.
At Hvidøre she lived very modest with a small loyal Russian retinue, consisting of Countess Mengden, Prince Dolgoruky and Prince Vyasemsky. I have also read that her Russian relative Princess Helena of Mecklenburg-Strelitz lived with her.