First off, congratulations on four months of being free of self-harm - that's a major accomplishment
Also, I remember when you spoke of your trying home life before and it's nice to hear that you are entering a living arrangement that sounds more comfortable for you.
Your plight definitely sounds challenging, but I think you can find a way around it. I can't really offer an opinion on whether or not your aunt would feel obligated to tell your foster mom about your self-harming, as I'm obviously not her, but it sounds as if you trust your aunt and I would like to believe that she would keep your secret if you were to explain how important it is to you.
As for
how to tell her, if you are more comfortable writing a letter, it may be a good idea to give it to her before she leaves for work, as she could read it there without the risk of your cousins seeing it. I realize you don't wish to bother her so early before she leaves to begin her day, but I think something so important would take priority in her mind.
I understand that having a deep conversation with your aunt about depression and self-harm isn't very appealing, but it's good that you recognize this may be necessary. I would suggest trying to look at it as a positive - it's encouraging when people are willing to listen and learn about subjects they are not well-versed in. This would also allow you to explain things to her your own way.
In any event, I wish you the best with this and I hope the move goes well