Is the questioner sinful for delaying the performance of the Fajr prayer until after sunrise because she did not set an alarm, relying on her usual early awakening, and will her prayer be counted?

You bear no sin, and your prayer is valid. You have fulfilled your obligation of prayer immediately upon waking. One who is asleep is excused and not held accountable, for the Prophet, peace and blessings be upon him, said: "There is no negligence in sleep." Your fear of sleeping alone is a pathological fear, and you must resist it and strive to overcome it.