Classmate shops my unpublished ideas around and contacts my advisors,
What you describe sounds very odd. It's hard to know exactly without being party to the conversations (and you don't know exactly what this person is saying to others about your work), but my initial impression is that this is more likely to hurt your friend than you. I also wonder if they're in a good space - it sounds less like someone trying to build a career and more like someone who might be a bit lost and confused about their own directions.
That said, it would absolutely be appropriate for you to ask that they stop talking about your work to third parties. It's one thing if it happens to come up in related conversation centered on their own work, it's another thing to shop it around actively, whether giving you credit or not. There are definitely kinder ways to phrase this and harsher ways. I'd probably start with kindness just because they are more likely to react positively. You can thank them for being excited about your research, but share that you'd like to spread your own work and ask that they not discuss it with people besides you.
You can also simply stop sharing such detail with them, without cutting them off completely unless you also choose to do that. If they are upset, I think it's reasonable to remind them that when you told them more they told other people, and that you'd rather talk about (any other topic). If they react aggressively, it may then make sense to get other people involved at your institution.
I'd also suggest having a conversation with your advisor(s). It doesn't need to be in any depth, and you don't need to set aside a separate meeting. Just mention that you've been talking about your research with a friend and you were surprised that they then conveyed it to third parties, and ask whether they think you should do anything about it. You don't have to accuse them or provide proof of plagiarism - you can just say what has happened and let others judge for themselves. They probably won't suggest there is much you can do, but now they'll be aware in the future.
Otherwise, keep moving forward and push towards publishing the things you have, like you would be doing anyways. If you have near-publishable work, consider discussing with your advisor whether it could be released as a pre-print. Whatever you have told this person about your progress so far, you have a huge head start on them, it's your research. Unless you think they'll actually be able to somehow beat you to publishing through their machinations, which doesn't seem likely from your description, they aren't too much of a threat.