im moving out to germany for my year abroad in september so should be able to help you out, unfortunately ive let the german get a bit rusty over the last month or so. "Hier schreibt Ihre Kusine Beth aus England" sounds pretty good, not really sure how else to put it. i think it is right to use 'Sie' yep, seeing as they are a distant relative. you could tell her to end it with 'Mit freundlichen Gruessen' or 'Herzliche Gruesse aus England', but what you have is equally good. any more questions just ask.
Did you cry ? Be honest
in a word, yes