Rawr! So sorry you ran into issues with your Bear. We do ask you to reach out to us at tunnelbear.com/support so that we can troubleshoot and investigate the issue. You CAN use the extension in incognito mode!
To clarify, when using a VPN it would not matter if you used incognito mode or not, all DNS requests would still need to go through our encrypted servers. That said, as per our strict Privacy and No-logging policy, TunnelBear would NEVER log these request. Please reach out to us for further information on this, and for troubleshooting so we can fix any issues!