Before reading what I have written, I would like to say that I am a non-native English speaker. Typo's, sentences that don't read like a train or nuances that might be a bit of may be the result.
Anyways; The things you have written about your cousin sounds like she has been (and thus probably will be) in this state of mind for some time now. This is a bad sign, because normally people will pull themselves together and move on. Now that she is dwelling on this state is a sign that she needs help from others to get past this. I would suggest, since you are her only outlet for now, to let her keep open up to you. If you cut off contact now you might make her feel more isolated and she will close even more to people... As well, it will create a new pasture to grow negative feelings on. However, this does not mean that you should solve her problems for her or not encourage her to do so. Try to encourage her to do things that are active. Sporting is a great way, because it will create chemicals that make you feel good. As well, it is good for psychical health and will improve her looks. Although shallow, it is a way that people can build self-esteem off. Discipline, pulling though are other things that come into this as well.
I would suggest not to steer her to solo-activities. Although it will give her something to do, she keeps isolated by these activities. If she want to make new friends, or develop her social skills, she has to interact with people in some way. -Internet- although fun, is not a replacement for the real world.
Two questions I would like to ask you:
What is her daily schedule? Does she go to school? If there isn't a routine, this is probably the first step you should work for. Give her a reason to get out of bed on time. Fill the day for her (or better, let her do it herself is she is able to), so that she has less time to think about things. As well, a structure and be able to predict what happens the next day gives a safe feeling. There is a reason why mental health wards have strict day programs.
Does she sleep on time? Does she sleeps alot or little? As a social worker I have noticed that sleeping is one huge fucking factor of how someone experience the quality of their lives. Compare it yourself, if you have a day after a horrible night of sleep, or waking rejuvenated. It makes a huge difference. Now think what would happen if you had a month of those horrible nights of sleep after each other. It would make me depressed, no doubt about that. So, if she has a unhealthy sleeping habit, tackle that down as well.
I would involve family by the way. You said you will be moving out in a while. Make sure there is someone who can keep an eye on her and support her when needed. The suggestions of activities (sporting, soup kitchen, learning life skills) are good and hopefully she will pursue them. But the changes are that she will have a relapse when you are gone (negative habits and thoughts will poke up, they always do) and then she will need someone to pull her through.