COM1&2 are most likely for onboard ports that don't have the required headers connected.

If you go to Device manager, and select the properties for your USB-RS232 adapter, it should show the COM port number somewhere within the properties (I would tell you exactly where, but I'm on my laptop and don't have any adapters to hand to try). Does that COM port show up within the Leadshine software?

If it doesn't you may need to close and re-open the Leadshine software so it refreshed the COM port list.

If it does show, it could be a cable problem.