Hi Sadye,
May I ask what your open rate is? I ask because a "good" open rate by industry standards is around 20%, give or take a few, and a good click through rate is around 3%, both of which seem low to many people.
At any rate, how targeted is your email list? Did everyone specifically opt-in to receive volunteer opportunity updates? If not, this might be why. Research shows email is still the best way to reach people but if they're not responding, it could be a number of reasons, such as: subject line isn't compelling enough, it's going to spam, they aren't available for volunteer opportunities right now, they don't remember signing up, they're just busy, etc.
I love your idea of doing a survey but I would simply do it to ask all of your volunteers how they'd like to hear about opportunities. (Not just specifically a survey to see if they use facebook. They may, but email still might be the best way to reach them, if done correctly.) Another option is instead of creating a new facebook group, have more people join the Core one? Also, if you're not already, you could post a link to your volunteer opportunities weekly on your organization page, and make sure you're reminding people to like your pages in emails.
Testing different days/times of day that you send your email is always smart, too.
Here's a blog post that might be of interest to you: http://chewonthis.maddiesfund.org/2017/05/are-your-emails-to-supporters-getting-deleted-or-ignored-heres-how-to-fix-that/
Best of luck to you!
#DataandTechnology