While all of those components are important, I really think it comes down to the fit of your research interests with those of the faculty at whatever schools you're applying to. It was always my dream to go to a feminist studies program, and I had such a hard time finding professors who could mentor me that I ended up accepting an anthropology offer over two GWSS offers (still a little bummed about it). It could be even harder to justify your case to a school when you're not using an explicit gender studies perspective - committees reject so many great people for any number of small reasons. You really have to sell them on the fact that YOU belong there over the many other SUPER QUALIFIED applicants.
Great for you that you found one professor already - I would say keep trying to personally connect with professors you're interested in working with, and also talk to current graduate students. Read the research that comes out of these departments and think about how yours fits in with it - are you using similar lenses/language, etc? I think a lot of people don't understand that gender studies isn't just about "women's issues" and you need to speak to the larger disciplinary POV and terminology. I did development studies in undergrad and I knew a ton of people who studied gender in one project and thought that qualified them to do a gender studies MA. (Not saying you're doing that, just to be mindful of it.) If you want to do gender studies over sociology, you really have to make a case for why that framework shapes your work.
Make sure you don't send the same statement of purpose to every school - tailor as many details of your application as you can to the specific program. That can be hard (I know, I applied 13 places this cycle), but I think sending in a handful of really thorough applications is better than just sending one general one everywhere.
Also - have as many people read your research paper as possible. Professors, colleagues, friends, etc. I was originally really proud of what I submitted (after having it reviewed by two former profs), but I read it recently and wanted to *facepalm* so much. There is always room to grow and add nuance to your arguments. Plus, even multiple rounds of editors can miss small grammatical errors. Same goes for you SoP - I ended up paying a professional editor to look at mine (I can be a little bit on the verbose side), and it really helped me craft a stronger and more succinct statement. It's easy to forget that the people reading our applications don't know us at all. Really use the space you have to knock them over the head with why you belong!
Good luck to you, feel free to PM me if you need anything else!
They'd be lucky to have you! You are a source of support and all around positive human being, I know that if you want a PhD, it will be there eventually. Thanks for being so cool to everyone this year.