How to search who exactly to cold-email for a research internship?
You're right that it's hard for an undergraduate to know where to start, and you've done the right thing by asking professors at your own university first.
The next thing to try would be to ask those professors again, but instead of asking to work with them, ask for their suggestions of a couple of people at other universities who you could get in touch with. You can do this iteratively: each time you email someone to ask for an opportunity, say that you understand if they're too busy to take on a student and ask if they have a suggestion for someone else you can contact.
Those more senior to me may weigh in on this, but it's generally frowned upon to offer your work for free. If the professor isn't able to compensate you financially, would you be able to get course credit for your work? I'm not too familiar with the US system, but perhaps it is possible. I would avoid working for free if you can -- it sets a precedent and allows professors to favour those who have their own financial support (and thus puts less well off students at a disadvantage when looking for research opportunities).
Keep studying the literature if you can, and persevere with emailing professors. To maximise your chances of a response, keep the email polite and to the point. Good luck!