Hi,
Yes, USG gateway WAN1 IP should be from the same network as RUTX09 LAN IP, in your case if RUTX09 has LAN IP 10.0.0.254, then you'd need to assign USG gateway WAN1 port interface IP address 10.0.0.1-253, if you're using static IP configuration, then you'd need to configure gateway IP (10.0.0.254) and DNS servers.
When you've tried to test your internet connection, have you tried to ping specific IP's e.g. 1.1.1.1 or 8.8.8.8 to determine if DNS servers aren't resolving hostnames or if you don't have internet access completely?