The refurbished /e/ phones are being shipped with Android 8, which does not natively support encrypted DNS. Nevertheless, NextDNS (an encrypted DNS provider) provides an app that allows you to use encrypted DNS with their servers on Android 8 phones.
Unfortunately, even though their app appears to claim that encrypted DNS is enabled, and even though there is an icon in the status bar showing that the app is running in the background, when I check what my DNS server is, it says that I am using my previous DNS server (which happened to be cloudflare). Even after restarting my phone (with NextDNS automatically starting), it still shows that I am using my old DNS server.
I opened a support ticket at NextDNS and they claim that the app should work fine on Android 8, and that therefore the problem is most likely related to /e/OS.