I've actually had to learn to develop my disclosure,,,
I've always had a belief that we fully open ourselves to others who are in certain roles - partner, therapist, etc. I have learned that if I am not more careful, the things I share can end up in the hands of folks who do not protect me in the way I anticipate. Whether this is a boyfriend who uses previously-shared intimate information in an argument with me or a therapist who doesn't know me deeply and grabs some segment of what I have shared and both misunderstands it and misplaces it in importance.
I am working on discernment, on sharing smaller bits of myself, observing how that information is received and handled, and then slowly opening in a dance with that reception. This is important for my recovery, because self-protection is something I have always been clumsy with - I don't self-protect early in a situation, and then blunder into a desperate move to self-protect "too-late."
In particular, I would use discernment as you slowly open up about the self-harm/suicidal piece. You are going to a low-income clinic, and it is likely that many of the practitioners there are advanced (graduate level) students working through their internship hours. In school they have been trained to immediately throw themselves toward the red-flag of self harm/suicide statements. They will feel the need to report it to their supervisor, to write it up in your notes, and perhaps to take further action to "protect you."
Unfortunately, even with medical privacy laws, there are some concerns that over-ride privacy. I have also seen medical notes taken on me after the fact, and much of what I thought I was "sharing privately" is summarized there for others to see, years after that intended private conversation with a therapist or doctor.
So, while I agree that you will receive the most accurate aid if you are able to be open about your deepest self, I think you should share that self at a pace in which you are gauging your safety and making actual decisions about how open you want to be in this situation. You might - for example - decide that you will share openly about your alcohol and drug use in the first session and then see how your therapist responds, see how you feel afterward (no regret at oversharing? a sense that you were truly heard? no sense that you were misinterpreted?). Then decide what you are willing to open up about at the next session.
Therapy/counselling is a relationship. You don't have to accomplish everything on the first date! Open, learn, evaluate, open a little more, and so on. It actually is like dating, that tidal opening and retreating, as you grow in intimacy.
I think that we addicts and alcoholics are most comfortable (and expect) immediate intimacy, and it is often too much for others to process and deal with. Slowing that down is a good thing (for many of us - there are also many who never shared their true self at all or revealed themselves or expressed emotional vulnerability, and those folks have a different pile o' issues to address).