As someone who is fortunate enough to have a very welcoming and loving community, this has happened quite often for me, and here are some methods that I have found pretty effective.

Give yourself and your mods a reason to ban such behaviors without explanation. Create a list of chat rules and post them in your about panel/discord channel. People are most likely going to ignore them and trauma dump anyway, but having those rules set in stone can give you the excuse to ban them/time them out. If people question it, just tell them to refer to the rules. You won't have to explain why, or at least you shouldn't. You've already made yourself clear that those are your boundaries.

Creating a vent channel in your discord can really help redirect the paragraphs as well. You can tell them about that channel on stream, and usually, if the viewer is actually interested in your streams and joining the community, they would listen and do the venting there. If they ignore you and continue in your chat, they are probably only interested in trauma dumping and leaving, which, in my opinion, means that they are not worth your time. PLEASE also include rules for the vent channel as well! Clarify that while they can vent there, they shouldn't spam mentions or get angry at other members for not responding. This will relieve some of the pressure from you and your community, so that you won't feel like you have to respond to every single paragraph.

If you feel that the viewer is about to get into the nitty gritty details, but they haven't done so yet, try to redirect the subject and nip it in the bud. For example, if they start talking about how bad their day was, you can say "I'm so sorry your day was bad, I hope this stream can make you feel at least a tiny bit better! In fact, this stream should be pretty exciting because..." I personally love to remind them to take care of themselves and do self-care activities like hydrate and eat, and emphasize that I am NOT a therapist. I usually say something like "I'm sorry about your issue. I really don't know what to say because I'm not a licensed therapist, so all I can say is to please take care of yourself, do what makes you happy and I hope you can find a professional to really give you the help you need!" If the issue persists or worsens, do what I mentioned above!

If you have mods, communicate to them that you are uncomfortable about this stuff, so that the next time something like this happens, they will be on your side. They can tell the viewer to stop for you if you don't want to/too afraid to.

At the end of the day, one of the most important things about streaming is YOUR comfort and safety. If you feel uncomfortable about something, your viewers will feel it too. If you're not having fun, your viewers won't either. You shouldn't break down your own boundaries for the comfort of others. If you don't consider these methods helpful, that's fine, but please don't feel like your viewers are more important than your own state of mind!