I've definitely done that, sent my T an email even though there's nothing she can really do. It usually goes something like, "I feel really really terrible. It's not an emergency; I don't need to go to the hospital. But I feel awful and hopeless [or whatever it is that I'm feeling] and I don't want to be alone in it."

Usually just sending her the email and knowing she'll read it makes me feel at least a tiny bit better. And she almost always responds (usually within 24 hours, sooner if it's in the evening before 10p or so), and says something caring and sympathetic, or gives me some practical suggestions (e.g. "could you try taking a warm bath?").

It helps. It doesn't magically fix everything, but when things are that bad it means a lot to me that she knows and she cares.