After some more troubleshooting, I was able to solve the issue.
For anyone who may stumble upon this in the future, the issue was that the <div> that enclosed the images and body copy of the signup was set to be locked at 500px. Media queries didn't appear to be working since this 500px width was set as an inline css style on the div itself. I removed that width property and created a new class (simply called .newsletter-signup-1) and adjusted the width at the desktop breakpoint to be 500px, and at the mobile breakpoint to be 275px.
After those adjustments, I was able to achieve a more properly sized box on mobile, image here.
Hope that helps!