The Surfshark WireGuard config files should already contain DNS IP address(es) that are within the U.S. :us: when you connect to their U.S. :us: servers. If not, the appropriate DNS IP address(es) would be assigned after you connect to their servers.

You should not have to manually set up Custom DNS Server in the GL.iNet router. The DNS IP addresses that Surfshark Support gave you may not work whenever you are not already connected to their servers and, hence, may be stopping you from connecting to the Internet.

Can you post one of their WireGuard config files, with personal information redacted?