Your therapist will not be shocked by what you say. Many people have suicidal thoughts. The therapist will try to evaluate with you how likely you are to act on the thoughts you are having now. If you don't seem likely to actually attempt suicide, the therapist will probably do nothing other than talk to you about the thoughts, in the context of the other problems you are experiencing. If you are likely to take action imminently, the therapist will send you for evaluation and possible admission to a psychiatric facility to keep you safe until medication and treatment can begin to have an effect. in this case, imminent means imminent: you have the razor in your hand, the supply of pills and the glass of water and so on. The therapist cannot tell your family without your permission.
I suggest that you tell the therapist about your thoughts early on in the session. Just be honest. You really have very little to fear.
Thirty years as a therapist