"so I have my VPN connection set to only route traffic through the VPN that goes to my company's resources."

First, be sure to ask your system administrator for security issues -- many companies do not like you to have both "normal" internet access and a VPN connection into their network active simultaniuosly (as this basically implies that any interactive malware/backdoor/trojan could access the company's network).

So: if you cannot browse the internet while the VPN connection is active (thus: if your administrator has disabled outgoing connections from the VPN gateway machine) then this might be on purpose!

And after that: go into your VPN settings, button "Advanced", "Session Options": uncheck "Send all traffic over VPN connection".