I wrote a tool that adds DNS support for ZeroTier’s Linux client: GitHub - twisteroidambassador/zerotier-resolved: Resolve hostnames for ZeroTier networks on Linux
It uses systemd-resolved to configure the DNS servers, so it should work on many desktop Linux distros. Once installed, it will automatically configure the system every time the ZeroTier network interface comes up.
Please try it out and see whether it works for you!
I wrote this after seeing GitHub - zerotier/zerotier-systemd-manager: Manages systemd per-interface DNS resolution for zeronsd . This one does the same thing, but requires systemd-network in addition to systemd-resolved, making it not as applicable to desktop distros.