Thanks for the responses, everyone.
It does seem like this could be a USB adapter issue. Here's the one I've been using, for anyone curious:
https://www.amazon.com/UCEC-Mouse-Keyboard-Converter-Cable/dp/B00X9QX9MS/ref=sr_1_2_sspa. Many reviews for that converter say that it worked for Model Ms, so I may want to purchase another (or try a different converter, like the Monoprice one mentioned) and see if that works.
Regarding under-powered USB ports: That shouldn't be an issue if the USB converter is plugged in directly to my computer (not through a USB hub), right?
The membrane is likely physically stuck together. You could try cleaning it, that'd probably fix the issue.
I plan on giving the keyboard a deeper cleaning sometime soon, and I can see if that changes anything. Based on how the board's behaving, though, it seems like this is more likely a USB converter issue.