I am writing this letter to invite you come to visit my new house.
.... As per the way your prompt sounds, this letter is not intended primarily for asking your friend to visit your home. It is to inform her that you moved to a new house. So, I guess you have given more prominence to what the prompt asks you last.
Also, she is your friend and therefore your letter can sound more intimacy.
My cousin is getting marrymarried this sunday and his family will move into my old house to start their marriage-life. Therefore, I have to move out.
.... This part is a bit confusing. This is my suggestion;
My cousin will be getting married on next Sunday and he hopes to settle down in the same house with his wife after the wedding. Since they are starting a new life I do not wish you disturb their privacy and hence thought I should move out.