You are not the problem, but the depression you have does affect the lives of those closest to you. Shutting people out is not good. Being a private person can be a curse, because you bottle stuff up that needs to be let out.
I speak from very sad experience as my FIL committed suicide 4 years ago, and subsequently my husband had severe depression and a nervous breakdown.
You cannot deal with it on your own, honestly.
Your sister is probably feeling frustrated, which is a normal reaction. It doesnt mean that she doesnt care. She wanted to come to the docs with you, because she cares and possibly also would like to know how she can help. She may also want to check that you are being honest about your feeling to your GP.
When my husband was ill I went through many emotions.. sympathy, frustration, leaving him to it, being on his case to get it sorted, and I also confided in other family members, because the last thing I wanted was for him to do what his father had done.
Let people in, let them help, talk to them honestly about how you are feeling. You may not always get the answers or responses that you think you should get, but you have to talk.