Traceroute won’t show if you leak DNS requests outside of your VPN. (Unless you coincidentally also leak traffic, but then you’re pretty much just not using your VPN).
To confirm you’ll need to analyze your traffic-flow using a tool like tcpdump or Wireshark and check the source and destination for DNS traffic. If you see incoming DNS responses on an interface that is not your VPN-adaptor or maybe a loopback interface then you’re probably not tunnelling DNS through the VPN.
To answer the question in the headline: Regular DNS is unencrypted and quite easy to snoop on, so any node on the route between you and the DNS server will be able to read it if not using a VPN (i.e. DNS leak). Not sure what you mean by adversary, but it’s not like anyone on the internet can see your traffic. The DNS server may log your request and if you’re not on VPN, your IP address may be logged too.