This may be because the verizon adapter is higher up in the binding order, so it will most likely look at that DNS server and if it gets a reply, it will go with that.
You can change the binding order, by opening going to the Network and Sharing Center, then going to Network Adapter settings. In that window, click on Alt, and a menu bar will appear. Click on Advanced -> Advanced Settings, then move the Cisco VPN client adapter up.
The other option you have is to use split-dns, and specify your internal DNS, or the domain names you want to lookup through your internal DNS servers supplied to you by the ASA. Split-dns is configured under the group-policy. Note that if you try to lookup a name that is supposed to be on the outside, and your DNS server can not reply with an answer, then the lookup will fail.
HTH
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