I had that problem using VPNBook, a free VPN service. I had my router configured as the DNS server. Being on the LAN, traffic to the router couldn't go through the VPN, obviously. Of course, the router was simply passing my DNS queries to Verizon.
When I realized what was going on, I modified the VPN startup script to switch DNS over to Google's public servers, at 8.8.8.8 and 8.8.4.4.
There are several sites that will check if you are leaking DNS queries. E.g., here is how dnsleaktest.com describes their service:
The DNS leak test works by sending your client a series of domain names to resolve within a specific test domain. Each request is sent from your client to your configured DNS server. Even if you have configured a single DNS server, there may be many other servers that the request is passed on to in order to be resolved (normally to load balance the requests). For example if you configure Google DNS then you will often find 6-10 Google DNS servers which are fullfilling the DNS requests.The Standard test performs 1 round of 6 queries for a total of 6 queries. This should be more than sufficient to discover if you have a DNS leak. The original test prior to the 2014 upgrade did 3 queries. The advantage of this test is that it is fast. Use this test to quickly check for dns leaks when you connect to your VPN service.
The Extended test performs 6 rounds of 6 queries for a total of 36 queries. This high number ensures that all DNS servers are discovered. However for the purposes of discovering if you have a DNS leak this is not normally neccessary and can take 10-30 seconds longer to complete. If you have strong anonymity/privacy requirements you can choose the extended test just to be certain.