After you have connected the VPN, wait a few seconds and then try loading
https://tunnelblick.net/ipinfo in a browser. (That's the URL Tunnelblick uses to check the IP address.)
If that works, there could be some kind of timing problem in the way that Tunnelblick is checking the IP address. Try changing the delay before the IP address check is made with the following Terminal command:
defaults write net.tunnelblick.tunnelblick delayBeforeIPAddressCheckAfterConnection -float 10.0
which changes the delay from 5 seconds (the default) to 10 seconds. If that work, you can try to lower it -- the minimum value is 0.1 seconds and the maximum is 180 seconds.
The reason for the delay is that OpenVPN usually (depending on your OpenVPN configuration) tells Tunnelblick that the VPN is "connected" before it is fully in place with all routing commands finished. The default delay of 5 seconds is usually long enough for everything to finish. Another approach would be to put a check in the "Set DNS after routes are set" checkbox in Tunnelblick's Advanced settings window. (Select the configuration(s) that you want the do that for before putting a check in the checkbox.)