It’s not completely clear what you mean, but I’m guessing you’re only routing a subset of your traffic through wireguard, probably only IPv4, and there may be some IPv6 traffic that is not being routed over your wireguard connection.

You can specify any IPs you want for DNS with wireguard, and if your allowed IPs include those addresses, then it should flow over your VPN.

I do this with Pihole at home, and it blocks ads while I’m away.

With whatever test you’re running that says stuff is “leaking,” keep in mind that the website is going to report any traffic that originates from your VPS as “unprotected” because it’s not their system, and even if you run your own DNS server, it’s still got to query upstream to a public DNS. All they’re really doing is demonstrating which upstream DNS server you have configured, and it’s up to you if you want your VPS’s IP to be connected to the query history of that upstream DNS provider.

You will usually need a hostname in DNS for your VPN server to make it easy to find/connect, which will use your normal DNS resolution. Once connected, if you have it set up correctly, new dns queries should route through your VPN connection. Just keep in mind that various results can be cached on your system and in web browsers, so you should quit and reopen your browser after you connect to the VPN before you run your “leak” test.