I would definitely suggest emailing the professors and asking, very politely, how you could improve to be a stronger candidate for the future/what your problem areas were. There are a number of issues that could have caused you not getting an acceptance, and I doubt it was your GRE score. While your quant is a bit low, I don't think it would play a major factor. Likely it had to do with research fit, your letters of rec, or your personal statement. Have you done any independent research? Do you have particular areas in mind that you want to further? We're you able to talk about your research experiences and what they have taught you? I think sometimes people may get rejected if they lack a certain depth--it's not enough to be interested in research; rather, you need to show you can think creatively and contribute to existing research
I think the research masters is a great idea, you would get more experience and hopefully more publications out of it as well. Additionally you could try to get a research position somewhere.