However, when I test what my ip address is from the tor browser, it returns the tor exit node address, not the airvpn server address. And when I try another browser, which is not proxied through tor, the ip address returned is that of the airvpn server address.
That's just fine. When OpenVPN is connected to Tor, and you use another application which is configured to connect to Tor, that application traffic will be tunneled over Tor only. You can see the reason by looking at the routing table and the gateways of your system. On a qualitative level, let's say that an application configured to connect to Tor proxy locally will connect to Tor proxy locally, it has no way to route the traffic on the tun interface to reach the VPN gateway which can be reached only over OpenVPN over Tor (OpenVPN connects to Tor proxy locally as well).
In https://airvpn.org/tor you can see the above depicted in the first picture.
If you need Tor over OpenVPN, connect OpenVPN directly to a VPN server, then use Tor. All the applications configured to connect to Tor will have their traffic tunneled over Tor over OpenVPN.