In Synology DSM, Control Panel → Network.
General Tab → Manually Configure DNS server.
Suggest using a public DNS, such as 1.1.1.1 (Cloudflare), 8.8.8.8/8.8.4.4 (Google), or 9.9.9.9 (Quad9).
After fixing DNS issues…
Synology FAQ 31: Reset / Reclaim PMS on DSM 7 (Not Authorized - Locked Out)
Uninstall Plex Media Server. After the “are you sure” page, select the “Uninstall Only” (first) option.
Reinstall the same version of Plex Media Server using the Plex Claim Token option. If successful, you will see “Plex Media Server successfully claimed by email address / userid.”