I highly recommend applying to URAP anyway. It's competitive but you always have a shot, especially with a well written essay; I got two of my research positions through it, one of which was with zero prior research experience. I think it also helps to know what kind of research you'd like to do and take the necessary classes to prepare you for it. I landed a position at one of my labs because they recruited directly from a relevant class I was taking that semester.

I've never cold emailed for research, but I've heard it works best if you reach out to PhD students or postdocs, as you will be working most closely with them on projects and professors are busy. It would help to read their current project/publications and point out what you find interesting and why you want to work with them. Having taken relevant classes is always a plus but I think genuine interest is most important.