Hi! What you are experiencing is not uncommon with those of us that suffer agoraphobia. Your cousin is your "safe" person and that's why it's easier to do things with her. We tend to feel more comfortable with certain individuals.
Try talking to your mother and sister about how you are feeling. Maybe then you can find the same level of comfort as with your cousin. Also, you may have some hidden personal issues with them that needs tending.
Are all these people aware of your situation? And how do they respond to it?
I know that I can go out with certain people and not others, too. For me, I have figured out it is because I know they truly understand what I am going through and will be compassionate if I panic versus make fun or belittle me. These individuals usually comfort me and talk to me until I am comfortable in what we are doing. They make me feel like they could take control if something happened...which in turn, makes me feel safer.
I wish you the best...I know hard this all is, as I have lived it on and off for many, many years. In honesty, the true answer is to find the safety within ourselves. :)